7085 lines
293 KiB
Plaintext
7085 lines
293 KiB
Plaintext
This is mpir.info, produced by makeinfo version 4.11 from mpir.texi.
|
||
|
||
This manual describes how to install and use MPIR, the Multiple
|
||
Precision Integers and Rationals library, version 1.0.
|
||
|
||
Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
|
||
2001, 2002, 2003, 2004, 2005, 2006 Free Software Foundation, Inc.
|
||
|
||
Copyright 2008 William Hart
|
||
|
||
Permission is granted to copy, distribute and/or modify this
|
||
document under the terms of the GNU Free Documentation License, Version
|
||
1.2 or any later version published by the Free Software Foundation;
|
||
with no Invariant Sections, with the Front-Cover Texts being "A GNU
|
||
Manual", and with the Back-Cover Texts being "You have freedom to copy
|
||
and modify this GNU Manual, like GNU software". A copy of the license
|
||
is included in *note GNU Free Documentation License::.
|
||
|
||
INFO-DIR-SECTION GNU libraries
|
||
START-INFO-DIR-ENTRY
|
||
* mpir: (mpir). MPIR Multiple Precision Integers and Rationals Library.
|
||
END-INFO-DIR-ENTRY
|
||
|
||
|
||
File: mpir.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
|
||
|
||
MPIR
|
||
****
|
||
|
||
This manual describes how to install and use MPIR, the Multiple
|
||
Precision Integers and Rationals library, version 1.0.
|
||
|
||
Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
|
||
2001, 2002, 2003, 2004, 2005, 2006 Free Software Foundation, Inc.
|
||
|
||
Copyright 2008 William Hart
|
||
|
||
Permission is granted to copy, distribute and/or modify this
|
||
document under the terms of the GNU Free Documentation License, Version
|
||
1.2 or any later version published by the Free Software Foundation;
|
||
with no Invariant Sections, with the Front-Cover Texts being "A GNU
|
||
Manual", and with the Back-Cover Texts being "You have freedom to copy
|
||
and modify this GNU Manual, like GNU software". A copy of the license
|
||
is included in *note GNU Free Documentation License::.
|
||
|
||
|
||
* Menu:
|
||
|
||
* Copying:: MPIR Copying Conditions (LGPL).
|
||
* Introduction to MPIR:: Brief introduction to MPIR.
|
||
* Installing MPIR:: How to configure and compile the MPIR library.
|
||
* MPIR Basics:: What every MPIR user should know.
|
||
* Reporting Bugs:: How to usefully report bugs.
|
||
* Integer Functions:: Functions for arithmetic on signed integers.
|
||
* Rational Number Functions:: Functions for arithmetic on rational numbers.
|
||
* Floating-point Functions:: Functions for arithmetic on floats.
|
||
* Low-level Functions:: Fast functions for natural numbers.
|
||
* Random Number Functions:: Functions for generating random numbers.
|
||
* Formatted Output:: `printf' style output.
|
||
* Formatted Input:: `scanf' style input.
|
||
* C++ Class Interface:: Class wrappers around MPIR types.
|
||
* BSD Compatible Functions:: All functions found in BSD MP.
|
||
* Custom Allocation:: How to customize the internal allocation.
|
||
* Language Bindings:: Using MPIR from other languages.
|
||
* Algorithms:: What happens behind the scenes.
|
||
* Internals:: How values are represented behind the scenes.
|
||
|
||
* Contributors:: Who brings you this library?
|
||
* References:: Some useful papers and books to read.
|
||
* GNU Free Documentation License::
|
||
* Concept Index::
|
||
* Function Index::
|
||
|
||
|
||
File: mpir.info, Node: Copying, Next: Introduction to MPIR, Prev: Top, Up: Top
|
||
|
||
MPIR Copying Conditions
|
||
***********************
|
||
|
||
This library is "free"; this means that everyone is free to use it and
|
||
free to redistribute it on a free basis. The library is not in the
|
||
public domain; it is copyrighted and there are restrictions on its
|
||
distribution, but these restrictions are designed to permit everything
|
||
that a good cooperating citizen would want to do. What is not allowed
|
||
is to try to prevent others from further sharing any version of this
|
||
library that they might get from you.
|
||
|
||
Specifically, we want to make sure that you have the right to give
|
||
away copies of the library, that you receive source code or else can
|
||
get it if you want it, that you can change this library or use pieces
|
||
of it in new free programs, and that you know you can do these things.
|
||
|
||
To make sure that everyone has such rights, we have to forbid you to
|
||
deprive anyone else of these rights. For example, if you distribute
|
||
copies of the MPIR library, you must give the recipients all the rights
|
||
that you have. You must make sure that they, too, receive or can get
|
||
the source code. And you must tell them their rights.
|
||
|
||
Also, for our own protection, we must make certain that everyone
|
||
finds out that there is no warranty for the MPIR library. If it is
|
||
modified by someone else and passed on, we want their recipients to
|
||
know that what they have is not what we distributed, so that any
|
||
problems introduced by others will not reflect on our reputation.
|
||
|
||
The precise conditions of the license for the MPIR library are found
|
||
in the Lesser General Public License version 2.1 that accompanies the
|
||
source code, see `COPYING.LIB'. Certain demonstration programs are
|
||
provided under the terms of the plain General Public License version 2,
|
||
see `COPYING'.
|
||
|
||
|
||
File: mpir.info, Node: Introduction to MPIR, Next: Installing MPIR, Prev: Copying, Up: Top
|
||
|
||
1 Introduction to MPIR
|
||
**********************
|
||
|
||
MPIR is a portable library written in C for arbitrary precision
|
||
arithmetic on integers, rational numbers, and floating-point numbers.
|
||
It aims to provide the fastest possible arithmetic for all applications
|
||
that need higher precision than is directly supported by the basic C
|
||
types.
|
||
|
||
Many applications use just a few hundred bits of precision; but some
|
||
applications may need thousands or even millions of bits. MPIR is
|
||
designed to give good performance for both, by choosing algorithms
|
||
based on the sizes of the operands, and by carefully keeping the
|
||
overhead at a minimum.
|
||
|
||
The speed of MPIR is achieved by using fullwords as the basic
|
||
arithmetic type, by using sophisticated algorithms, by including
|
||
carefully optimized assembly code for the most common inner loops for
|
||
many different CPUs, and by a general emphasis on speed (as opposed to
|
||
simplicity or elegance).
|
||
|
||
There is assembly code for these CPUs: ARM, DEC Alpha 21064, 21164,
|
||
and 21264, AMD 29000, AMD K6, K6-2, Athlon, K8 and K10, Hitachi SuperH
|
||
and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium Pro/II/III,
|
||
Pentium 4, generic x86, Intel IA-64, i960, Core 2 Motorola MC68000,
|
||
MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64, National
|
||
NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic
|
||
SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some optimizations also
|
||
for Cray vector systems, Clipper, IBM ROMP (RT), and Pyramid AP/XP.
|
||
|
||
For up-to-date information on, and latest version of, MPIR, please see
|
||
the MPIR web pages at
|
||
|
||
`http://www.mpir.org/'
|
||
|
||
There are a number of public mailing lists of interest. The
|
||
development list is
|
||
|
||
`http://groups.google.com/group/mpir-devel/'.
|
||
|
||
The proper place for bug reports is
|
||
`http://groups.google.com/group/mpir-devel'. See *note Reporting
|
||
Bugs:: for information about reporting bugs.
|
||
|
||
|
||
1.1 How to use this Manual
|
||
==========================
|
||
|
||
Everyone should read *note MPIR Basics::. If you need to install the
|
||
library yourself, then read *note Installing MPIR::. If you have a
|
||
system with multiple ABIs, then read *note ABI and ISA::, for the
|
||
compiler options that must be used on applications.
|
||
|
||
The rest of the manual can be used for later reference, although it
|
||
is probably a good idea to glance through it.
|
||
|
||
|
||
File: mpir.info, Node: Installing MPIR, Next: MPIR Basics, Prev: Introduction to MPIR, Up: Top
|
||
|
||
2 Installing MPIR
|
||
*****************
|
||
|
||
MPIR has an autoconf/automake/libtool based configuration system. On a
|
||
Unix-like system a basic build can be done with
|
||
|
||
./configure
|
||
make
|
||
|
||
Some self-tests can be run with
|
||
|
||
make check
|
||
|
||
And you can install (under `/usr/local' by default) with
|
||
|
||
make install
|
||
|
||
Important note: by default MPIR produces libraries named libmpir,
|
||
etc., and the header file mpir.h. If you wish to have MPIR to install
|
||
as a library named libgmp as well, etc., and a gmp.h header file, so
|
||
that you can use mpir with programs designed to only work with GMP,
|
||
then use the `--enable-gmpcompat' option when invoking configure:
|
||
|
||
./configure --enable-gmpcompat
|
||
|
||
If you experience problems, please report them to
|
||
`http://groups.google.com/group/mpir-devel'. See *note Reporting
|
||
Bugs::, for information on what to include in useful bug reports.
|
||
|
||
* Menu:
|
||
|
||
* Build Options::
|
||
* ABI and ISA::
|
||
* Notes for Package Builds::
|
||
* Notes for Particular Systems::
|
||
* Known Build Problems::
|
||
* Performance optimization::
|
||
|
||
|
||
File: mpir.info, Node: Build Options, Next: ABI and ISA, Prev: Installing MPIR, Up: Installing MPIR
|
||
|
||
2.1 Build Options
|
||
=================
|
||
|
||
All the usual autoconf configure options are available, run `./configure
|
||
--help' for a summary. The file `INSTALL.autoconf' has some generic
|
||
installation information too.
|
||
|
||
Tools
|
||
`configure' requires various Unix-like tools. See *note Notes for
|
||
Particular Systems::, for some options on non-Unix systems.
|
||
|
||
It might be possible to build without the help of `configure',
|
||
certainly all the code is there, but unfortunately you'll be on
|
||
your own.
|
||
|
||
Build Directory
|
||
To compile in a separate build directory, `cd' to that directory,
|
||
and prefix the configure command with the path to the MPIR source
|
||
directory. For example
|
||
|
||
cd /my/build/dir
|
||
/my/sources/mpir-1.0/configure
|
||
|
||
Not all `make' programs have the necessary features (`VPATH') to
|
||
support this. In particular, SunOS and Solaris `make' have bugs
|
||
that make them unable to build in a separate directory. Use GNU
|
||
`make' instead.
|
||
|
||
`--prefix' and `--exec-prefix'
|
||
The `--prefix' option can be used in the normal way to direct MPIR
|
||
to install under a particular tree. The default is `/usr/local'.
|
||
|
||
`--exec-prefix' can be used to direct architecture-dependent files
|
||
like `libmpir.a' to a different location. This can be used to
|
||
share architecture-independent parts like the documentation, but
|
||
separate the dependent parts. Note however that `mpir.h' and
|
||
`mp.h' are architecture-dependent since they encode certain
|
||
aspects of `libmpir', so it will be necessary to ensure both
|
||
`$prefix/include' and `$exec_prefix/include' are available to the
|
||
compiler.
|
||
|
||
`--enable-gmpcompat'
|
||
By default make builds libmpir library files (and libmpirxx if C++
|
||
headers are requested) and the mpir.h header file. This option
|
||
allows you to specify that you want additional libraries created
|
||
called libgmp (and libgmpxx), etc., for libraries and gmp.h for
|
||
compatibility with GMP.
|
||
|
||
`--disable-shared', `--disable-static'
|
||
By default both shared and static libraries are built (where
|
||
possible), but one or other can be disabled. Shared libraries
|
||
result in smaller executables and permit code sharing between
|
||
separate running processes, but on some CPUs are slightly slower,
|
||
having a small cost on each function call.
|
||
|
||
Native Compilation, `--build=CPU-VENDOR-OS'
|
||
For normal native compilation, the system can be specified with
|
||
`--build'. By default `./configure' uses the output from running
|
||
`./config.guess'. On some systems `./config.guess' can determine
|
||
the exact CPU type, on others it will be necessary to give it
|
||
explicitly. For example,
|
||
|
||
./configure --build=ultrasparc-sun-solaris2.7
|
||
|
||
In all cases the `OS' part is important, since it controls how
|
||
libtool generates shared libraries. Running `./config.guess' is
|
||
the simplest way to see what it should be, if you don't know
|
||
already.
|
||
|
||
Cross Compilation, `--host=CPU-VENDOR-OS'
|
||
When cross-compiling, the system used for compiling is given by
|
||
`--build' and the system where the library will run is given by
|
||
`--host'. For example when using a FreeBSD Athlon system to build
|
||
GNU/Linux m68k binaries,
|
||
|
||
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
|
||
|
||
Compiler tools are sought first with the host system type as a
|
||
prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then
|
||
plain `ranlib'. This makes it possible for a set of
|
||
cross-compiling tools to co-exist with native tools. The prefix
|
||
is the argument to `--host', and this can be an alias, such as
|
||
`m68k-linux'. But note that tools don't have to be setup this
|
||
way, it's enough to just have a `PATH' with a suitable
|
||
cross-compiling `cc' etc.
|
||
|
||
Compiling for a different CPU in the same family as the build
|
||
system is a form of cross-compilation, though very possibly this
|
||
would merely be special options on a native compiler. In any case
|
||
`./configure' avoids depending on being able to run code on the
|
||
build system, which is important when creating binaries for a
|
||
newer CPU since they very possibly won't run on the build system.
|
||
|
||
In all cases the compiler must be able to produce an executable
|
||
(of whatever format) from a standard C `main'. Although only
|
||
object files will go to make up `libmpir', `./configure' uses
|
||
linking tests for various purposes, such as determining what
|
||
functions are available on the host system.
|
||
|
||
Currently a warning is given unless an explicit `--build' is used
|
||
when cross-compiling, because it may not be possible to correctly
|
||
guess the build system type if the `PATH' has only a
|
||
cross-compiling `cc'.
|
||
|
||
Note that the `--target' option is not appropriate for MPIR. It's
|
||
for use when building compiler tools, with `--host' being where
|
||
they will run, and `--target' what they'll produce code for.
|
||
Ordinary programs or libraries like MPIR are only interested in
|
||
the `--host' part, being where they'll run.
|
||
|
||
CPU types
|
||
In general, if you want a library that runs as fast as possible,
|
||
you should configure MPIR for the exact CPU type your system uses.
|
||
However, this may mean the binaries won't run on older members of
|
||
the family, and might run slower on other members, older or newer.
|
||
The best idea is always to build MPIR for the exact machine type
|
||
you intend to run it on.
|
||
|
||
The following CPUs have specific support. See `configure.in' for
|
||
details of what code and compiler options they select.
|
||
|
||
* Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
|
||
alphaev6, alphaev67, alphaev68 alphaev7
|
||
|
||
* Cray: c90, j90, t90, sv1
|
||
|
||
* HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w, hppa64
|
||
|
||
* IA-64: ia64, itanium, itanium2
|
||
|
||
* MIPS: mips, mips3, mips64
|
||
|
||
* Motorola: m68k, m68000, m68010, m68020, m68030, m68040,
|
||
m68060, m68302, m68360, m88k, m88110
|
||
|
||
* POWER: power, power1, power2, power2sc
|
||
|
||
* PowerPC: powerpc, powerpc64, powerpc401, powerpc403,
|
||
powerpc405, powerpc505, powerpc601, powerpc602, powerpc603,
|
||
powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630,
|
||
powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801,
|
||
powerpc821, powerpc823, powerpc860, powerpc970
|
||
|
||
* SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9,
|
||
ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64
|
||
|
||
* x86 family: pentium, pentiummmx, pentiumpro, pentium2,
|
||
pentium3, pentium4, netburst, netburstlahf, prescott, core,
|
||
core2, penryn, nehalem, atom, k5, k6, k62, k63, k7, k8, k10
|
||
viac3, viac32
|
||
|
||
* Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax,
|
||
z8k
|
||
|
||
CPUs not listed will use generic C code.
|
||
|
||
Generic C Build
|
||
If some of the assembly code causes problems, or if otherwise
|
||
desired, the generic C code can be selected with CPU `none'. For
|
||
example,
|
||
|
||
./configure --host=none-unknown-freebsd3.5
|
||
|
||
Note that this will run quite slowly, but it should be portable
|
||
and should at least make it possible to get something running if
|
||
all else fails.
|
||
|
||
Fat binary, `--enable-fat'
|
||
Using `--enable-fat' selects a "fat binary" build on x86 or x86_64
|
||
systems, where optimized low level subroutines are chosen at
|
||
runtime according to the CPU detected. This means more code, but
|
||
gives reasonable performance from a single binary for all x86
|
||
chips, or simililarly for all x86_64 chips. (This option might
|
||
become available for more architectures in the future.)
|
||
|
||
`ABI'
|
||
On some systems MPIR supports multiple ABIs (application binary
|
||
interfaces), meaning data type sizes and calling conventions. By
|
||
default MPIR chooses the best ABI available, but a particular ABI
|
||
can be selected. For example
|
||
|
||
./configure --host=mips64-sgi-irix6 ABI=n32
|
||
|
||
See *note ABI and ISA::, for the available choices on relevant
|
||
CPUs, and what applications need to do.
|
||
|
||
`CC', `CFLAGS'
|
||
By default the C compiler used is chosen from among some likely
|
||
candidates, with `gcc' normally preferred if it's present. The
|
||
usual `CC=whatever' can be passed to `./configure' to choose
|
||
something different.
|
||
|
||
For various systems, default compiler flags are set based on the
|
||
CPU and compiler. The usual `CFLAGS="-whatever"' can be passed to
|
||
`./configure' to use something different or to set good flags for
|
||
systems MPIR doesn't otherwise know.
|
||
|
||
The `CC' and `CFLAGS' used are printed during `./configure', and
|
||
can be found in each generated `Makefile'. This is the easiest way
|
||
to check the defaults when considering changing or adding
|
||
something.
|
||
|
||
Note that when `CC' and `CFLAGS' are specified on a system
|
||
supporting multiple ABIs it's important to give an explicit
|
||
`ABI=whatever', since MPIR can't determine the ABI just from the
|
||
flags and won't be able to select the correct assembler code.
|
||
|
||
If just `CC' is selected then normal default `CFLAGS' for that
|
||
compiler will be used (if MPIR recognises it). For example
|
||
`CC=gcc' can be used to force the use of GCC, with default flags
|
||
(and default ABI).
|
||
|
||
`CPPFLAGS'
|
||
Any flags like `-D' defines or `-I' includes required by the
|
||
preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'.
|
||
Compiling is done with both `CPPFLAGS' and `CFLAGS', but
|
||
preprocessing uses just `CPPFLAGS'. This distinction is because
|
||
most preprocessors won't accept all the flags the compiler does.
|
||
Preprocessing is done separately in some configure tests, and in
|
||
the `ansi2knr' support for K&R compilers.
|
||
|
||
`CC_FOR_BUILD'
|
||
Some build-time programs are compiled and run to generate
|
||
host-specific data tables. `CC_FOR_BUILD' is the compiler used
|
||
for this. It doesn't need to be in any particular ABI or mode, it
|
||
merely needs to generate executables that can run. The default is
|
||
to try the selected `CC' and some likely candidates such as `cc'
|
||
and `gcc', looking for something that works.
|
||
|
||
No flags are used with `CC_FOR_BUILD' because a simple invocation
|
||
like `cc foo.c' should be enough. If some particular options are
|
||
required they can be included as for instance `CC_FOR_BUILD="cc
|
||
-whatever"'.
|
||
|
||
C++ Support, `--enable-cxx'
|
||
C++ support in MPIR can be enabled with `--enable-cxx', in which
|
||
case a C++ compiler will be required. As a convenience
|
||
`--enable-cxx=detect' can be used to enable C++ support only if a
|
||
compiler can be found. The C++ support consists of a library
|
||
`libmpirxx.la' and header file `mpirxx.h' (*note Headers and
|
||
Libraries::).
|
||
|
||
A separate `libmpirxx.la' has been adopted rather than having C++
|
||
objects within `libmpir.la' in order to ensure dynamic linked C
|
||
programs aren't bloated by a dependency on the C++ standard
|
||
library, and to avoid any chance that the C++ compiler could be
|
||
required when linking plain C programs.
|
||
|
||
`libmpirxx.la' will use certain internals from `libmpir.la' and can
|
||
only be expected to work with `libmpir.la' from the same MPIR
|
||
version. Future changes to the relevant internals will be
|
||
accompanied by renaming, so a mismatch will cause unresolved
|
||
symbols rather than perhaps mysterious misbehaviour.
|
||
|
||
In general `libmpirxx.la' will be usable only with the C++
|
||
compiler that built it, since name mangling and runtime support
|
||
are usually incompatible between different compilers.
|
||
|
||
`CXX', `CXXFLAGS'
|
||
When C++ support is enabled, the C++ compiler and its flags can be
|
||
set with variables `CXX' and `CXXFLAGS' in the usual way. The
|
||
default for `CXX' is the first compiler that works from a list of
|
||
likely candidates, with `g++' normally preferred when available.
|
||
The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without
|
||
`-g', then for `g++' either `-g -O2' or `-O2', or for other
|
||
compilers `-g' or nothing. Trying `CFLAGS' this way is convenient
|
||
when using `gcc' and `g++' together, since the flags for `gcc' will
|
||
usually suit `g++'.
|
||
|
||
It's important that the C and C++ compilers match, meaning their
|
||
startup and runtime support routines are compatible and that they
|
||
generate code in the same ABI (if there's a choice of ABIs on the
|
||
system). `./configure' isn't currently able to check these things
|
||
very well itself, so for that reason `--disable-cxx' is the
|
||
default, to avoid a build failure due to a compiler mismatch.
|
||
Perhaps this will change in the future.
|
||
|
||
Incidentally, it's normally not good enough to set `CXX' to the
|
||
same as `CC'. Although `gcc' for instance recognises `foo.cc' as
|
||
C++ code, only `g++' will invoke the linker the right way when
|
||
building an executable or shared library from C++ object files.
|
||
|
||
Temporary Memory, `--enable-alloca=<choice>'
|
||
MPIR allocates temporary workspace using one of the following
|
||
three methods, which can be selected with for instance
|
||
`--enable-alloca=malloc-reentrant'.
|
||
|
||
* `alloca' - C library or compiler builtin.
|
||
|
||
* `malloc-reentrant' - the heap, in a re-entrant fashion.
|
||
|
||
* `malloc-notreentrant' - the heap, with global variables.
|
||
|
||
For convenience, the following choices are also available.
|
||
`--disable-alloca' is the same as `no'.
|
||
|
||
* `yes' - a synonym for `alloca'.
|
||
|
||
* `no' - a synonym for `malloc-reentrant'.
|
||
|
||
* `reentrant' - `alloca' if available, otherwise
|
||
`malloc-reentrant'. This is the default.
|
||
|
||
* `notreentrant' - `alloca' if available, otherwise
|
||
`malloc-notreentrant'.
|
||
|
||
`alloca' is reentrant and fast, and is recommended. It actually
|
||
allocates just small blocks on the stack; larger ones use
|
||
malloc-reentrant.
|
||
|
||
`malloc-reentrant' is, as the name suggests, reentrant and thread
|
||
safe, but `malloc-notreentrant' is faster and should be used if
|
||
reentrancy is not required.
|
||
|
||
The two malloc methods in fact use the memory allocation functions
|
||
selected by `mp_set_memory_functions', these being `malloc' and
|
||
friends by default. *Note Custom Allocation::.
|
||
|
||
An additional choice `--enable-alloca=debug' is available, to help
|
||
when debugging memory related problems (*note Debugging::).
|
||
|
||
FFT Multiplication, `--disable-fft'
|
||
By default multiplications are done using Karatsuba, 3-way Toom,
|
||
and Fermat FFT. The FFT is only used on large to very large
|
||
operands and can be disabled to save code size if desired.
|
||
|
||
Berkeley MP, `--enable-mpbsd'
|
||
The Berkeley MP compatibility library (`libmp') and header file
|
||
(`mp.h') are built and installed only if `--enable-mpbsd' is used.
|
||
*Note BSD Compatible Functions::.
|
||
|
||
Assertion Checking, `--enable-assert'
|
||
This option enables some consistency checking within the library.
|
||
This can be of use while debugging, *note Debugging::.
|
||
|
||
Execution Profiling, `--enable-profiling=prof/gprof/instrument'
|
||
Enable profiling support, in one of various styles, *note
|
||
Profiling::.
|
||
|
||
`MPN_PATH'
|
||
Various assembler versions of each mpn subroutines are provided.
|
||
For a given CPU, a search is made though a path to choose a
|
||
version of each. For example `sparcv8' has
|
||
|
||
MPN_PATH="sparc32/v8 sparc32 generic"
|
||
|
||
which means look first for v8 code, then plain sparc32 (which is
|
||
v7), and finally fall back on generic C. Knowledgeable users with
|
||
special requirements can specify a different path. Normally this
|
||
is completely unnecessary.
|
||
|
||
Documentation
|
||
The source for the document you're now reading is `doc/mpir.texi',
|
||
in Texinfo format, see *note Texinfo: (texinfo)Top.
|
||
|
||
Info format `doc/mpir.info' is included in the distribution. The
|
||
usual automake targets are available to make PostScript, DVI, PDF
|
||
and HTML (these will require various TeX and Texinfo tools).
|
||
|
||
DocBook and XML can be generated by the Texinfo `makeinfo' program
|
||
too, see *note Options for `makeinfo': (texinfo)makeinfo options.
|
||
|
||
Some supplementary notes can also be found in the `doc'
|
||
subdirectory.
|
||
|
||
|
||
|
||
File: mpir.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing MPIR
|
||
|
||
2.2 ABI and ISA
|
||
===============
|
||
|
||
ABI (Application Binary Interface) refers to the calling conventions
|
||
between functions, meaning what registers are used and what sizes the
|
||
various C data types are. ISA (Instruction Set Architecture) refers to
|
||
the instructions and registers a CPU has available.
|
||
|
||
Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI
|
||
defined, the latter for compatibility with older CPUs in the family.
|
||
MPIR supports some CPUs like this in both ABIs. In fact within MPIR
|
||
`ABI' means a combination of chip ABI, plus how MPIR chooses to use it.
|
||
For example in some 32-bit ABIs, MPIR may support a limb as either a
|
||
32-bit `long' or a 64-bit `long long'.
|
||
|
||
By default MPIR chooses the best ABI available for a given system,
|
||
and this generally gives significantly greater speed. But an ABI can
|
||
be chosen explicitly to make MPIR compatible with other libraries, or
|
||
particular application requirements. For example,
|
||
|
||
./configure ABI=32
|
||
|
||
In all cases it's vital that all object code used in a given program
|
||
is compiled for the same ABI.
|
||
|
||
Usually a limb is implemented as a `long'. When a `long long' limb
|
||
is used this is encoded in the generated `mpir.h'. This is convenient
|
||
for applications, but it does mean that `mpir.h' will vary, and can't
|
||
be just copied around. `mpir.h' remains compiler independent though,
|
||
since all compilers for a particular ABI will be expected to use the
|
||
same limb type.
|
||
|
||
Currently no attempt is made to follow whatever conventions a system
|
||
has for installing library or header files built for a particular ABI.
|
||
This will probably only matter when installing multiple builds of MPIR,
|
||
and it might be as simple as configuring with a special `libdir', or it
|
||
might require more than that. Note that builds for different ABIs need
|
||
to done separately, with a fresh `./configure' and `make' each.
|
||
|
||
|
||
AMD64 (`x86_64')
|
||
On AMD64 systems supporting both 32-bit and 64-bit modes for
|
||
applications, the following ABI choices are available.
|
||
|
||
`ABI=64'
|
||
The 64-bit ABI uses 64-bit limbs and pointers and makes full
|
||
use of the chip architecture. This is the default.
|
||
Applications will usually not need special compiler flags,
|
||
but for reference the option is
|
||
|
||
gcc -m64
|
||
|
||
`ABI=32'
|
||
The 32-bit ABI is the usual i386 conventions. This will be
|
||
slower, and is not recommended except for inter-operating
|
||
with other code not yet 64-bit capable. Applications must be
|
||
compiled with
|
||
|
||
gcc -m32
|
||
|
||
(In GCC 2.95 and earlier there's no `-m32' option, it's the
|
||
only mode.)
|
||
|
||
|
||
HPPA 2.0 (`hppa2.0*', `hppa64')
|
||
|
||
`ABI=2.0w'
|
||
The 2.0w ABI uses 64-bit limbs and pointers and is available
|
||
on HP-UX 11 or up. Applications must be compiled with
|
||
|
||
gcc [built for 2.0w]
|
||
cc +DD64
|
||
|
||
`ABI=2.0n'
|
||
The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal
|
||
calling conventions, but with 64-bit instructions permitted
|
||
within functions. MPIR uses a 64-bit `long long' for a limb.
|
||
This ABI is available on hppa64 GNU/Linux and on HP-UX 10 or
|
||
higher. Applications must be compiled with
|
||
|
||
gcc [built for 2.0n]
|
||
cc +DA2.0 +e
|
||
|
||
Note that current versions of GCC (eg. 3.2) don't generate
|
||
64-bit instructions for `long long' operations and so may be
|
||
slower than for 2.0w. (The MPIR assembler code is the same
|
||
though.)
|
||
|
||
`ABI=1.0'
|
||
HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit
|
||
HPPA 1.0 ABI. No special compiler options are needed for
|
||
applications.
|
||
|
||
All three ABIs are available for CPU types `hppa2.0w', `hppa2.0'
|
||
and `hppa64', but for CPU type `hppa2.0n' only 2.0n or 1.0 are
|
||
considered.
|
||
|
||
Note that GCC on HP-UX has no options to choose between 2.0n and
|
||
2.0w modes, unlike HP `cc'. Instead it must be built for one or
|
||
the other ABI. MPIR will detect how it was built, and skip to the
|
||
corresponding `ABI'.
|
||
|
||
|
||
IA-64 under HP-UX (`ia64*-*-hpux*', `itanium*-*-hpux*')
|
||
HP-UX supports two ABIs for IA-64. MPIR performance is the same
|
||
in both.
|
||
|
||
`ABI=32'
|
||
In the 32-bit ABI, pointers, `int's and `long's are 32 bits
|
||
and MPIR uses a 64 bit `long long' for a limb. Applications
|
||
can be compiled without any special flags since this ABI is
|
||
the default in both HP C and GCC, but for reference the flags
|
||
are
|
||
|
||
gcc -milp32
|
||
cc +DD32
|
||
|
||
`ABI=64'
|
||
In the 64-bit ABI, `long's and pointers are 64 bits and MPIR
|
||
uses a `long' for a limb. Applications must be compiled with
|
||
|
||
gcc -mlp64
|
||
cc +DD64
|
||
|
||
On other IA-64 systems, GNU/Linux for instance, `ABI=64' is the
|
||
only choice.
|
||
|
||
|
||
MIPS under IRIX 6 (`mips*-*-irix[6789]')
|
||
IRIX 6 always has a 64-bit MIPS 3 or better CPU, and supports ABIs
|
||
o32, n32, and 64. n32 or 64 are recommended, and MPIR performance
|
||
will be the same in each. The default is n32.
|
||
|
||
`ABI=o32'
|
||
The o32 ABI is 32-bit pointers and integers, and no 64-bit
|
||
operations. MPIR will be slower than in n32 or 64, this
|
||
option only exists to support old compilers, eg. GCC 2.7.2.
|
||
Applications can be compiled with no special flags on an old
|
||
compiler, or on a newer compiler with
|
||
|
||
gcc -mabi=32
|
||
cc -32
|
||
|
||
`ABI=n32'
|
||
The n32 ABI is 32-bit pointers and integers, but with a
|
||
64-bit limb using a `long long'. Applications must be
|
||
compiled with
|
||
|
||
gcc -mabi=n32
|
||
cc -n32
|
||
|
||
`ABI=64'
|
||
The 64-bit ABI is 64-bit pointers and integers. Applications
|
||
must be compiled with
|
||
|
||
gcc -mabi=64
|
||
cc -64
|
||
|
||
Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have
|
||
the necessary support for n32 or 64 and so only gets a 32-bit limb
|
||
and the MIPS 2 code.
|
||
|
||
|
||
PowerPC 64 (`powerpc64', `powerpc620', `powerpc630', `powerpc970')
|
||
|
||
`ABI=aix64'
|
||
The AIX 64 ABI uses 64-bit limbs and pointers and is the
|
||
default on PowerPC 64 `*-*-aix*' systems. Applications must
|
||
be compiled with
|
||
|
||
gcc -maix64
|
||
xlc -q64
|
||
|
||
`ABI=mode32'
|
||
The `mode32' ABI uses a 64-bit `long long' limb but with the
|
||
chip still in 32-bit mode and using 32-bit calling
|
||
conventions. This is the default on PowerPC 64 `*-*-darwin*'
|
||
systems. No special compiler options are needed for
|
||
applications.
|
||
|
||
`ABI=32'
|
||
This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No
|
||
special compiler options are needed for applications.
|
||
|
||
MPIR speed is greatest in `aix64' and `mode32'. In `ABI=32' only
|
||
the 32-bit ISA is used and this doesn't make full use of a 64-bit
|
||
chip. On a suitable system we could perhaps use more of the ISA,
|
||
but there are no plans to do so.
|
||
|
||
|
||
Sparc V9 (`sparc64', `sparcv9', `ultrasparc*')
|
||
|
||
`ABI=64'
|
||
The 64-bit V9 ABI is available on the various BSD sparc64
|
||
ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7
|
||
and up (when the kernel is in 64-bit mode). GCC 3.2 or
|
||
higher, or Sun `cc' is required. On GNU/Linux, depending on
|
||
the default `gcc' mode, applications must be compiled with
|
||
|
||
gcc -m64
|
||
|
||
On Solaris applications must be compiled with
|
||
|
||
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
|
||
cc -xarch=v9
|
||
|
||
On the BSD sparc64 systems no special options are required,
|
||
since 64-bits is the only ABI available.
|
||
|
||
`ABI=32'
|
||
For the basic 32-bit ABI, MPIR still uses as much of the V9
|
||
ISA as it can. In the Sun documentation this combination is
|
||
known as "v8plus". On GNU/Linux, depending on the default
|
||
`gcc' mode, applications may need to be compiled with
|
||
|
||
gcc -m32
|
||
|
||
On Solaris, no special compiler options are required for
|
||
applications, though using something like the following is
|
||
recommended. (`gcc' 2.8 and earlier only support `-mv8'
|
||
though.)
|
||
|
||
gcc -mv8plus
|
||
cc -xarch=v8plus
|
||
|
||
MPIR speed is greatest in `ABI=64', so it's the default where
|
||
available. The speed is partly because there are extra registers
|
||
available and partly because 64-bits is considered the more
|
||
important case and has therefore had better code written for it.
|
||
|
||
Don't be confused by the names of the `-m' and `-x' compiler
|
||
options, they're called `arch' but effectively control both ABI
|
||
and ISA.
|
||
|
||
On Solaris 2.6 and earlier, only `ABI=32' is available since the
|
||
kernel doesn't save all registers.
|
||
|
||
On Solaris 2.7 with the kernel in 32-bit mode, a normal native
|
||
build will reject `ABI=64' because the resulting executables won't
|
||
run. `ABI=64' can still be built if desired by making it look
|
||
like a cross-compile, for example
|
||
|
||
./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
|
||
|
||
|
||
File: mpir.info, Node: Notes for Package Builds, Next: Notes for Particular Systems, Prev: ABI and ISA, Up: Installing MPIR
|
||
|
||
2.3 Notes for Package Builds
|
||
============================
|
||
|
||
MPIR should present no great difficulties for packaging in a binary
|
||
distribution.
|
||
|
||
Libtool is used to build the library and `-version-info' is set
|
||
appropriately, having started from `3:0:0' in GMP 3.0 (*note Library
|
||
interface versions: (libtool)Versioning.).
|
||
|
||
The GMP 4 series and MPIR 1 series will be upwardly binary
|
||
compatible in each release and will be upwardly binary compatible with
|
||
all of the GMP 3 series. Additional function interfaces may be added
|
||
in each release, so on systems where libtool versioning is not fully
|
||
checked by the loader an auxiliary mechanism may be needed to express
|
||
that a dynamic linked application depends on a new enough MPIR.
|
||
|
||
An auxiliary mechanism may also be needed to express that
|
||
`libmpirxx.la' (from `--enable-cxx', *note Build Options::) requires
|
||
`libmpir.la' from the same MPIR version, since this is not done by the
|
||
libtool versioning, nor otherwise. A mismatch will result in
|
||
unresolved symbols from the linker, or perhaps the loader.
|
||
|
||
When building a package for a CPU family, care should be taken to use
|
||
`--host' (or `--build') to choose the least common denominator among
|
||
the CPUs which might use the package. For example this might mean plain
|
||
`sparc' (meaning V7) for SPARCs.
|
||
|
||
For x86s, `--enable-fat' sets things up for a fat binary build,
|
||
making a runtime selection of optimized low level routines. This is a
|
||
good choice for packaging to run on a range of x86 chips.
|
||
|
||
Users who care about speed will want MPIR built for their exact CPU
|
||
type, to make best use of the available optimizations. Providing a way
|
||
to suitably rebuild a package may be useful. This could be as simple
|
||
as making it possible for a user to omit `--build' (and `--host') so
|
||
`./config.guess' will detect the CPU. But a way to manually specify a
|
||
`--build' will be wanted for systems where `./config.guess' is inexact.
|
||
|
||
On systems with multiple ABIs, a packaged build will need to decide
|
||
which among the choices is to be provided, see *note ABI and ISA::. A
|
||
given run of `./configure' etc will only build one ABI. If a second
|
||
ABI is also required then a second run of `./configure' etc must be
|
||
made, starting from a clean directory tree (`make distclean').
|
||
|
||
As noted under "ABI and ISA", currently no attempt is made to follow
|
||
system conventions for install locations that vary with ABI, such as
|
||
`/usr/lib/sparcv9' for `ABI=64' as opposed to `/usr/lib' for `ABI=32'.
|
||
A package build can override `libdir' and other standard variables as
|
||
necessary.
|
||
|
||
Note that `mpir.h' is a generated file, and will be architecture and
|
||
ABI dependent. When attempting to install two ABIs simultaneously it
|
||
will be important that an application compile gets the correct `mpir.h'
|
||
for its desired ABI. If compiler include paths don't vary with ABI
|
||
options then it might be necessary to create a `/usr/include/mpir.h'
|
||
which tests preprocessor symbols and chooses the correct actual
|
||
`mpir.h'.
|
||
|
||
|
||
File: mpir.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing MPIR
|
||
|
||
2.4 Notes for Particular Systems
|
||
================================
|
||
|
||
AIX 3 and 4
|
||
On systems `*-*-aix[34]*' shared libraries are disabled by
|
||
default, since some versions of the native `ar' fail on the
|
||
convenience libraries used. A shared build can be attempted with
|
||
|
||
./configure --enable-shared --disable-static
|
||
|
||
Note that the `--disable-static' is necessary because in a shared
|
||
build libtool makes `libmpir.a' a symlink to `libmpir.so',
|
||
apparently for the benefit of old versions of `ld' which only
|
||
recognise `.a', but unfortunately this is done even if a fully
|
||
functional `ld' is available.
|
||
|
||
ARM
|
||
On systems `arm*-*-*', versions of GCC up to and including 2.95.3
|
||
have a bug in unsigned division, giving wrong results for some
|
||
operands. MPIR `./configure' will demand GCC 2.95.4 or later.
|
||
|
||
Compaq C++
|
||
Compaq C++ on OSF 5.1 has two flavours of `iostream', a standard
|
||
one and an old pre-standard one (see `man iostream_intro'). MPIR
|
||
can only use the standard one, which unfortunately is not the
|
||
default but must be selected by defining `__USE_STD_IOSTREAM'.
|
||
Configure with for instance
|
||
|
||
./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
|
||
|
||
Floating Point Mode
|
||
On some systems, the hardware floating point has a control mode
|
||
which can set all operations to be done in a particular precision,
|
||
for instance single, double or extended on x86 systems (x87
|
||
floating point). The MPIR functions involving a `double' cannot
|
||
be expected to operate to their full precision when the hardware
|
||
is in single precision mode. Of course this affects all code,
|
||
including application code, not just MPIR.
|
||
|
||
MS-DOS and MS Windows
|
||
On an MS-DOS system DJGPP can be used to build MPIR, and on an MS
|
||
Windows system Cygwin, DJGPP and MINGW can all be used. All three
|
||
are excellent ports of GCC and the various GNU tools.
|
||
|
||
`http://www.cygwin.com/'
|
||
`http://www.delorie.com/djgpp/'
|
||
`http://www.mingw.org/'
|
||
|
||
In addition, project files for MSVC are provided, allowing MPIR to
|
||
build on Microsoft's compiler. This support is provided by Brian
|
||
Gladman.
|
||
|
||
Microsoft also publishes an Interix "Services for Unix" which can
|
||
be used to build MPIR on Windows (with a normal `./configure'),
|
||
but it's not free software.
|
||
|
||
MS Windows DLLs
|
||
On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default
|
||
MPIR builds only a static library, but a DLL can be built instead
|
||
using
|
||
|
||
./configure --disable-static --enable-shared
|
||
|
||
Static and DLL libraries can't both be built, since certain export
|
||
directives in `mpir.h' must be different.
|
||
|
||
A MINGW DLL build of MPIR can be used with Microsoft C. Libtool
|
||
doesn't install a `.lib' format import library, but it can be
|
||
created with MS `lib' as follows, and copied to the install
|
||
directory. Similarly for `libmp' and `libmpirxx'.
|
||
|
||
cd .libs
|
||
lib /def:libgmp-3.dll.def /out:libgmp-3.lib
|
||
|
||
MINGW uses the C runtime library `msvcrt.dll' for I/O, so
|
||
applications wanting to use the MPIR I/O routines must be compiled
|
||
with `cl /MD' to do the same. If one of the other C runtime
|
||
library choices provided by MS C is desired then the suggestion is
|
||
to use the MPIR string functions and confine I/O to the
|
||
application.
|
||
|
||
Motorola 68k CPU Types
|
||
`m68k' is taken to mean 68000. `m68020' or higher will give a
|
||
performance boost on applicable CPUs. `m68360' can be used for
|
||
CPU32 series chips. `m68302' can be used for "Dragonball" series
|
||
chips, though this is merely a synonym for `m68000'.
|
||
|
||
OpenBSD 2.6
|
||
`m4' in this release of OpenBSD has a bug in `eval' that makes it
|
||
unsuitable for `.asm' file processing. `./configure' will detect
|
||
the problem and either abort or choose another m4 in the `PATH'.
|
||
The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
|
||
|
||
Power CPU Types
|
||
In MPIR, CPU types `power*' and `powerpc*' will each use
|
||
instructions not available on the other, so it's important to
|
||
choose the right one for the CPU that will be used. Currently
|
||
MPIR has no assembler code support for using just the common
|
||
instruction subset. To get executables that run on both, the
|
||
current suggestion is to use the generic C code (CPU `none'),
|
||
possibly with appropriate compiler options (like `-mcpu=common' for
|
||
`gcc'). CPU `rs6000' (which is not a CPU but a family of
|
||
workstations) is accepted by `config.sub', but is currently
|
||
equivalent to `none'.
|
||
|
||
Sparc CPU Types
|
||
`sparcv8' or `supersparc' on relevant systems will give a
|
||
significant performance increase over the V7 code selected by plain
|
||
`sparc'.
|
||
|
||
Sparc App Regs
|
||
The MPIR assembler code for both 32-bit and 64-bit Sparc clobbers
|
||
the "application registers" `g2', `g3' and `g4', the same way that
|
||
the GCC default `-mapp-regs' does (*note SPARC Options: (gcc)SPARC
|
||
Options.).
|
||
|
||
This makes that code unsuitable for use with the special V9
|
||
`-mcmodel=embmedany' (which uses `g4' as a data segment pointer),
|
||
and for applications wanting to use those registers for special
|
||
purposes. In these cases the only suggestion currently is to
|
||
build MPIR with CPU `none' to avoid the assembler code.
|
||
|
||
SunOS 4
|
||
`/usr/bin/m4' lacks various features needed to process `.asm'
|
||
files, and instead `./configure' will automatically use
|
||
`/usr/5bin/m4', which we believe is always available (if not then
|
||
use GNU m4).
|
||
|
||
x86 CPU Types
|
||
`i586', `pentium' or `pentiummmx' code is good for its intended P5
|
||
Pentium chips, but quite slow when run on Intel P6 class chips
|
||
(PPro, P-II, P-III). `i386' is a better choice when making
|
||
binaries that must run on both.
|
||
|
||
x86 MMX and SSE2 Code
|
||
If the CPU selected has MMX code but the assembler doesn't support
|
||
it, a warning is given and non-MMX code is used instead. This
|
||
will be an inferior build, since the MMX code that's present is
|
||
there because it's faster than the corresponding plain integer
|
||
code. The same applies to SSE2.
|
||
|
||
Old versions of `gas' don't support MMX instructions, in particular
|
||
version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent
|
||
OpenBSD 3.1 doesn't.
|
||
|
||
Solaris 2.6 and 2.7 `as' generate incorrect object code for
|
||
register to register `movq' instructions, and so can't be used for
|
||
MMX code. Install a recent `gas' if MMX code is wanted on these
|
||
systems.
|
||
|
||
|
||
File: mpir.info, Node: Known Build Problems, Next: Performance optimization, Prev: Notes for Particular Systems, Up: Installing MPIR
|
||
|
||
2.5 Known Build Problems
|
||
========================
|
||
|
||
You might find more up-to-date information at `http://www.mpir.org/'.
|
||
|
||
Compiler link options
|
||
The version of libtool currently in use rather aggressively strips
|
||
compiler options when linking a shared library. This will
|
||
hopefully be relaxed in the future, but for now if this is a
|
||
problem the suggestion is to create a little script to hide them,
|
||
and for instance configure with
|
||
|
||
./configure CC=gcc-with-my-options
|
||
|
||
DJGPP (`*-*-msdosdjgpp*')
|
||
The DJGPP port of `bash' 2.03 is unable to run the `configure'
|
||
script, it exits silently, having died writing a preamble to
|
||
`config.log'. Use `bash' 2.04 or higher.
|
||
|
||
`make all' was found to run out of memory during the final
|
||
`libgmp.la' link on one system tested, despite having 64Mb
|
||
available. Running `make libgmp.la' directly helped, perhaps
|
||
recursing into the various subdirectories uses up memory.
|
||
|
||
GNU binutils `strip' prior to 2.12
|
||
`strip' from GNU binutils 2.11 and earlier should not be used on
|
||
the static libraries `libmpir.a' and `libmp.a' since it will
|
||
discard all but the last of multiple archive members with the same
|
||
name, like the three versions of `init.o' in `libmpir.a'.
|
||
Binutils 2.12 or higher can be used successfully.
|
||
|
||
The shared libraries `libmpir.so' and `libmp.so' are not affected
|
||
by this and any version of `strip' can be used on them.
|
||
|
||
`make' syntax error
|
||
On certain versions of SCO OpenServer 5 and IRIX 6.5 the native
|
||
`make' is unable to handle the long dependencies list for
|
||
`libmpir.la'. The symptom is a "syntax error" on the following
|
||
line of the top-level `Makefile'.
|
||
|
||
libmpir.la: $(libmpir_la_OBJECTS) $(libmpir_la_DEPENDENCIES)
|
||
|
||
Either use GNU Make, or as a workaround remove
|
||
`$(libmpir_la_DEPENDENCIES)' from that line (which will make the
|
||
initial build work, but if any recompiling is done `libmpir.la'
|
||
might not be rebuilt).
|
||
|
||
MacOS X (`*-*-darwin*')
|
||
Libtool currently only knows how to create shared libraries on
|
||
MacOS X using the native `cc' (which is a modified GCC), not a
|
||
plain GCC. A static-only build should work though
|
||
(`--disable-shared').
|
||
|
||
NeXT prior to 3.3
|
||
The system compiler on old versions of NeXT was a massacred and
|
||
old GCC, even if it called itself `cc'. This compiler cannot be
|
||
used to build MPIR, you need to get a real GCC, and install that.
|
||
(NeXT may have fixed this in release 3.3 of their system.)
|
||
|
||
POWER and PowerPC
|
||
Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile
|
||
MPIR on POWER or PowerPC. If you want to use GCC for these
|
||
machines, get GCC 2.7.2.1 (or later).
|
||
|
||
Sequent Symmetry
|
||
Use the GNU assembler instead of the system assembler, since the
|
||
latter has serious bugs.
|
||
|
||
Solaris 2.6
|
||
The system `sed' prints an error "Output line too long" when
|
||
libtool builds `libmpir.la'. This doesn't seem to cause any
|
||
obvious ill effects, but GNU `sed' is recommended, to avoid any
|
||
doubt.
|
||
|
||
Sparc Solaris 2.7 with gcc 2.95.2 in `ABI=32'
|
||
A shared library build of MPIR seems to fail in this combination,
|
||
it builds but then fails the tests, apparently due to some
|
||
incorrect data relocations within `gmp_randinit_lc_2exp_size'.
|
||
The exact cause is unknown, `--disable-shared' is recommended.
|
||
|
||
|
||
File: mpir.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing MPIR
|
||
|
||
2.6 Performance optimization
|
||
============================
|
||
|
||
For optimal performance, build MPIR for the exact CPU type of the target
|
||
computer, see *note Build Options::.
|
||
|
||
Unlike what is the case for most other programs, the compiler
|
||
typically doesn't matter much, since MPIR uses assembly language for
|
||
the most critical operation.
|
||
|
||
In particular for long-running MPIR applications, and applications
|
||
demanding extremely large numbers, building and running the `tuneup'
|
||
program in the `tune' subdirectory, can be important. For example,
|
||
|
||
cd tune
|
||
make tuneup
|
||
./tuneup
|
||
|
||
will generate better contents for the `gmp-mparam.h' parameter file.
|
||
|
||
To use the results, put the output in the file indicated in the
|
||
`Parameters for ...' header. Then recompile from scratch.
|
||
|
||
The `tuneup' program takes one useful parameter, `-f NNN', which
|
||
instructs the program how long to check FFT multiply parameters. If
|
||
you're going to use MPIR for extremely large numbers, you may want to
|
||
run `tuneup' with a large NNN value.
|
||
|
||
|
||
File: mpir.info, Node: MPIR Basics, Next: Reporting Bugs, Prev: Installing MPIR, Up: Top
|
||
|
||
3 MPIR Basics
|
||
*************
|
||
|
||
*Using functions, macros, data types, etc. not documented in this
|
||
manual is strongly discouraged. If you do so your application is
|
||
guaranteed to be incompatible with future versions of MPIR.*
|
||
|
||
* Menu:
|
||
|
||
* Headers and Libraries::
|
||
* Nomenclature and Types::
|
||
* Function Classes::
|
||
* Variable Conventions::
|
||
* Parameter Conventions::
|
||
* Memory Management::
|
||
* Reentrancy::
|
||
* Useful Macros and Constants::
|
||
* Compatibility with older versions::
|
||
* Demonstration Programs::
|
||
* Efficiency::
|
||
* Debugging::
|
||
* Profiling::
|
||
* Autoconf::
|
||
* Emacs::
|
||
|
||
|
||
File: mpir.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPIR Basics, Up: MPIR Basics
|
||
|
||
3.1 Headers and Libraries
|
||
=========================
|
||
|
||
All declarations needed to use MPIR are collected in the include file
|
||
`mpir.h'. It is designed to work with both C and C++ compilers.
|
||
|
||
#include <mpir.h>
|
||
|
||
Note however that prototypes for MPIR functions with `FILE *'
|
||
parameters are only provided if `<stdio.h>' is included too.
|
||
|
||
#include <stdio.h>
|
||
#include <mpir.h>
|
||
|
||
Likewise `<stdarg.h>' (or `<varargs.h>') is required for prototypes
|
||
with `va_list' parameters, such as `gmp_vprintf'. And `<obstack.h>'
|
||
for prototypes with `struct obstack' parameters, such as
|
||
`gmp_obstack_printf', when available.
|
||
|
||
All programs using MPIR must link against the `libmpir' library. On
|
||
a typical Unix-like system this can be done with `-lmpir' respectively,
|
||
for example
|
||
|
||
gcc myprogram.c -lmpir
|
||
|
||
MPIR C++ functions are in a separate `libmpirxx' library. This is
|
||
built and installed if C++ support has been enabled (*note Build
|
||
Options::). For example,
|
||
|
||
g++ mycxxprog.cc -lmpirxx -lmpir
|
||
|
||
MPIR is built using Libtool and an application can use that to link
|
||
if desired, *note GNU Libtool: (libtool)Top.
|
||
|
||
If MPIR has been installed to a non-standard location then it may be
|
||
necessary to use `-I' and `-L' compiler options to point to the right
|
||
directories, and some sort of run-time path for a shared library.
|
||
|
||
|
||
File: mpir.info, Node: Nomenclature and Types, Next: Function Classes, Prev: Headers and Libraries, Up: MPIR Basics
|
||
|
||
3.2 Nomenclature and Types
|
||
==========================
|
||
|
||
In this manual, "integer" usually means a multiple precision integer, as
|
||
defined by the MPIR library. The C data type for such integers is
|
||
`mpz_t'. Here are some examples of how to declare such integers:
|
||
|
||
mpz_t sum;
|
||
|
||
struct foo { mpz_t x, y; };
|
||
|
||
mpz_t vec[20];
|
||
|
||
"Rational number" means a multiple precision fraction. The C data
|
||
type for these fractions is `mpq_t'. For example:
|
||
|
||
mpq_t quotient;
|
||
|
||
"Floating point number" or "Float" for short, is an arbitrary
|
||
precision mantissa with a limited precision exponent. The C data type
|
||
for such objects is `mpf_t'. For example:
|
||
|
||
mpf_t fp;
|
||
|
||
The floating point functions accept and return exponents in the C
|
||
type `mp_exp_t'. Currently this is usually a `long', but on some
|
||
systems it's an `int' for efficiency.
|
||
|
||
A "limb" means the part of a multi-precision number that fits in a
|
||
single machine word. (We chose this word because a limb of the human
|
||
body is analogous to a digit, only larger, and containing several
|
||
digits.) Normally a limb is 32 or 64 bits. The C data type for a limb
|
||
is `mp_limb_t'.
|
||
|
||
Counts of limbs are represented in the C type `mp_size_t'. Currently
|
||
this is normally a `long', but on some systems it's an `int' for
|
||
efficiency.
|
||
|
||
"Random state" means an algorithm selection and current state data.
|
||
The C data type for such objects is `gmp_randstate_t'. For example:
|
||
|
||
gmp_randstate_t rstate;
|
||
|
||
Also, in general `unsigned long' is used for bit counts and ranges,
|
||
and `size_t' is used for byte or character counts.
|
||
|
||
|
||
File: mpir.info, Node: Function Classes, Next: Variable Conventions, Prev: Nomenclature and Types, Up: MPIR Basics
|
||
|
||
3.3 Function Classes
|
||
====================
|
||
|
||
There are six classes of functions in the MPIR library:
|
||
|
||
1. Functions for signed integer arithmetic, with names beginning with
|
||
`mpz_'. The associated type is `mpz_t'. There are about 150
|
||
functions in this class. (*note Integer Functions::)
|
||
|
||
2. Functions for rational number arithmetic, with names beginning with
|
||
`mpq_'. The associated type is `mpq_t'. There are about 40
|
||
functions in this class, but the integer functions can be used for
|
||
arithmetic on the numerator and denominator separately. (*note
|
||
Rational Number Functions::)
|
||
|
||
3. Functions for floating-point arithmetic, with names beginning with
|
||
`mpf_'. The associated type is `mpf_t'. There are about 60
|
||
functions is this class. (*note Floating-point Functions::)
|
||
|
||
4. Functions compatible with Berkeley MP, such as `itom', `madd', and
|
||
`mult'. The associated type is `MINT'. (*note BSD Compatible
|
||
Functions::)
|
||
|
||
5. Fast low-level functions that operate on natural numbers. These
|
||
are used by the functions in the preceding groups, and you can
|
||
also call them directly from very time-critical user programs.
|
||
These functions' names begin with `mpn_'. The associated type is
|
||
array of `mp_limb_t'. There are about 30 (hard-to-use) functions
|
||
in this class. (*note Low-level Functions::)
|
||
|
||
6. Miscellaneous functions. Functions for setting up custom
|
||
allocation and functions for generating random numbers. (*note
|
||
Custom Allocation::, and *note Random Number Functions::)
|
||
|
||
|
||
File: mpir.info, Node: Variable Conventions, Next: Parameter Conventions, Prev: Function Classes, Up: MPIR Basics
|
||
|
||
3.4 Variable Conventions
|
||
========================
|
||
|
||
MPIR functions generally have output arguments before input arguments.
|
||
This notation is by analogy with the assignment operator. The BSD MP
|
||
compatibility functions are exceptions, having the output arguments
|
||
last.
|
||
|
||
MPIR lets you use the same variable for both input and output in one
|
||
call. For example, the main function for integer multiplication,
|
||
`mpz_mul', can be used to square `x' and put the result back in `x' with
|
||
|
||
mpz_mul (x, x, x);
|
||
|
||
Before you can assign to an MPIR variable, you need to initialize it
|
||
by calling one of the special initialization functions. When you're
|
||
done with a variable, you need to clear it out, using one of the
|
||
functions for that purpose. Which function to use depends on the type
|
||
of variable. See the chapters on integer functions, rational number
|
||
functions, and floating-point functions for details.
|
||
|
||
A variable should only be initialized once, or at least cleared
|
||
between each initialization. After a variable has been initialized, it
|
||
may be assigned to any number of times.
|
||
|
||
For efficiency reasons, avoid excessive initializing and clearing.
|
||
In general, initialize near the start of a function and clear near the
|
||
end. For example,
|
||
|
||
void
|
||
foo (void)
|
||
{
|
||
mpz_t n;
|
||
int i;
|
||
mpz_init (n);
|
||
for (i = 1; i < 100; i++)
|
||
{
|
||
mpz_mul (n, ...);
|
||
mpz_fdiv_q (n, ...);
|
||
...
|
||
}
|
||
mpz_clear (n);
|
||
}
|
||
|
||
|
||
File: mpir.info, Node: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: MPIR Basics
|
||
|
||
3.5 Parameter Conventions
|
||
=========================
|
||
|
||
When an MPIR variable is used as a function parameter, it's effectively
|
||
a call-by-reference, meaning if the function stores a value there it
|
||
will change the original in the caller. Parameters which are
|
||
input-only can be designated `const' to provoke a compiler error or
|
||
warning on attempting to modify them.
|
||
|
||
When a function is going to return an MPIR result, it should
|
||
designate a parameter that it sets, like the library functions do.
|
||
More than one value can be returned by having more than one output
|
||
parameter, again like the library functions. A `return' of an `mpz_t'
|
||
etc doesn't return the object, only a pointer, and this is almost
|
||
certainly not what's wanted.
|
||
|
||
Here's an example accepting an `mpz_t' parameter, doing a
|
||
calculation, and storing the result to the indicated parameter.
|
||
|
||
void
|
||
foo (mpz_t result, const mpz_t param, unsigned long n)
|
||
{
|
||
unsigned long i;
|
||
mpz_mul_ui (result, param, n);
|
||
for (i = 1; i < n; i++)
|
||
mpz_add_ui (result, result, i*7);
|
||
}
|
||
|
||
int
|
||
main (void)
|
||
{
|
||
mpz_t r, n;
|
||
mpz_init (r);
|
||
mpz_init_set_str (n, "123456", 0);
|
||
foo (r, n, 20L);
|
||
gmp_printf ("%Zd\n", r);
|
||
return 0;
|
||
}
|
||
|
||
`foo' works even if the mainline passes the same variable for
|
||
`param' and `result', just like the library functions. But sometimes
|
||
it's tricky to make that work, and an application might not want to
|
||
bother supporting that sort of thing.
|
||
|
||
For interest, the MPIR types `mpz_t' etc are implemented as
|
||
one-element arrays of certain structures. This is why declaring a
|
||
variable creates an object with the fields MPIR needs, but then using
|
||
it as a parameter passes a pointer to the object. Note that the actual
|
||
fields in each `mpz_t' etc are for internal use only and should not be
|
||
accessed directly by code that expects to be compatible with future
|
||
MPIR releases.
|
||
|
||
|
||
File: mpir.info, Node: Memory Management, Next: Reentrancy, Prev: Parameter Conventions, Up: MPIR Basics
|
||
|
||
3.6 Memory Management
|
||
=====================
|
||
|
||
The MPIR types like `mpz_t' are small, containing only a couple of
|
||
sizes, and pointers to allocated data. Once a variable is initialized,
|
||
MPIR takes care of all space allocation. Additional space is allocated
|
||
whenever a variable doesn't have enough.
|
||
|
||
`mpz_t' and `mpq_t' variables never reduce their allocated space.
|
||
Normally this is the best policy, since it avoids frequent reallocation.
|
||
Applications that need to return memory to the heap at some particular
|
||
point can use `mpz_realloc2', or clear variables no longer needed.
|
||
|
||
`mpf_t' variables, in the current implementation, use a fixed amount
|
||
of space, determined by the chosen precision and allocated at
|
||
initialization, so their size doesn't change.
|
||
|
||
All memory is allocated using `malloc' and friends by default, but
|
||
this can be changed, see *note Custom Allocation::. Temporary memory
|
||
on the stack is also used (via `alloca'), but this can be changed at
|
||
build-time if desired, see *note Build Options::.
|
||
|
||
|
||
File: mpir.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: MPIR Basics
|
||
|
||
3.7 Reentrancy
|
||
==============
|
||
|
||
MPIR is reentrant and thread-safe, with some exceptions:
|
||
|
||
* If configured with `--enable-alloca=malloc-notreentrant' (or with
|
||
`--enable-alloca=notreentrant' when `alloca' is not available),
|
||
then naturally MPIR is not reentrant.
|
||
|
||
* `mpf_set_default_prec' and `mpf_init' use a global variable for the
|
||
selected precision. `mpf_init2' can be used instead, and in the
|
||
C++ interface an explicit precision to the `mpf_class' constructor.
|
||
|
||
* `mpz_random' and the other old random number functions use a global
|
||
random state and are hence not reentrant. The newer random number
|
||
functions that accept a `gmp_randstate_t' parameter can be used
|
||
instead.
|
||
|
||
* `gmp_randinit' (obsolete) returns an error indication through a
|
||
global variable, which is not thread safe. Applications are
|
||
advised to use `gmp_randinit_default' or `gmp_randinit_lc_2exp'
|
||
instead.
|
||
|
||
* `mp_set_memory_functions' uses global variables to store the
|
||
selected memory allocation functions.
|
||
|
||
* If the memory allocation functions set by a call to
|
||
`mp_set_memory_functions' (or `malloc' and friends by default) are
|
||
not reentrant, then MPIR will not be reentrant either.
|
||
|
||
* If the standard I/O functions such as `fwrite' are not reentrant
|
||
then the MPIR I/O functions using them will not be reentrant
|
||
either.
|
||
|
||
* It's safe for two threads to read from the same MPIR variable
|
||
simultaneously, but it's not safe for one to read while the
|
||
another might be writing, nor for two threads to write
|
||
simultaneously. It's not safe for two threads to generate a
|
||
random number from the same `gmp_randstate_t' simultaneously,
|
||
since this involves an update of that variable.
|
||
|
||
|
||
File: mpir.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: MPIR Basics
|
||
|
||
3.8 Useful Macros and Constants
|
||
===============================
|
||
|
||
-- Global Constant: const int mp_bits_per_limb
|
||
The number of bits per limb.
|
||
|
||
-- Macro: __GNU_MP_VERSION
|
||
-- Macro: __GNU_MP_VERSION_MINOR
|
||
-- Macro: __GNU_MP_VERSION_PATCHLEVEL
|
||
The major and minor GMP version, and patch level, respectively, as
|
||
integers. For GMP i.j, these numbers will be i, j, and 0,
|
||
respectively. For GMP i.j.k, these numbers will be i, j, and k,
|
||
respectively. These numbers represent the version of GMP fully
|
||
supported by this version of MPIR.
|
||
|
||
-- Macro: __MPIR_VERSION
|
||
-- Macro: __MPIR_VERSION_MINOR
|
||
-- Macro: __MPIR_VERSION_PATCHLEVEL
|
||
The major and minor MPIR version, and patch level, respectively,
|
||
as integers. For MPIR i.j, these numbers will be i, j, and 0,
|
||
respectively. For MPIR i.j.k, these numbers will be i, j, and k,
|
||
respectively.
|
||
|
||
-- Global Constant: const char * const gmp_version
|
||
The GNU MP version number, as a null-terminated string, in the
|
||
form "i.j" or "i.j.k".
|
||
|
||
-- Global Constant: const char * const mpir_version
|
||
The MPIR version number, as a null-terminated string, in the form
|
||
"i.j" or "i.j.k". This release is "1.0".
|
||
|
||
|
||
File: mpir.info, Node: Compatibility with older versions, Next: Demonstration Programs, Prev: Useful Macros and Constants, Up: MPIR Basics
|
||
|
||
3.9 Compatibility with older versions
|
||
=====================================
|
||
|
||
This version of MPIR is upwardly binary compatible with all GMP 4.x and
|
||
3.x versions, and upwardly compatible at the source level with all 2.x
|
||
versions, with the following exceptions.
|
||
|
||
* `mpn_gcd' had its source arguments swapped as of GMP 3.0, for
|
||
consistency with other `mpn' functions.
|
||
|
||
* `mpf_get_prec' counted precision slightly differently in GMP 3.0
|
||
and 3.0.1, but in 3.1 reverted to the 2.x style.
|
||
|
||
There are a number of compatibility issues between GMP 1 and GMP 2
|
||
that of course also apply when porting applications from GMP 1 to GMP 4
|
||
and MPIR 1. Please see the GMP 2 manual for details.
|
||
|
||
The Berkeley MP compatibility library (*note BSD Compatible
|
||
Functions::) is source and binary compatible with the standard `libmp'.
|
||
|
||
|
||
File: mpir.info, Node: Demonstration Programs, Next: Efficiency, Prev: Compatibility with older versions, Up: MPIR Basics
|
||
|
||
3.10 Demonstration programs
|
||
===========================
|
||
|
||
The `demos' subdirectory has some sample programs using MPIR. These
|
||
aren't built or installed, but there's a `Makefile' with rules for them.
|
||
For instance,
|
||
|
||
make pexpr
|
||
./pexpr 68^975+10
|
||
|
||
The following programs are provided
|
||
|
||
* `pexpr' is an expression evaluator, the program used on the GMP
|
||
web page.
|
||
|
||
* The `calc' subdirectory has a similar but simpler evaluator using
|
||
`lex' and `yacc'.
|
||
|
||
* The `expr' subdirectory is yet another expression evaluator, a
|
||
library designed for ease of use within a C program. See
|
||
`demos/expr/README' for more information.
|
||
|
||
* `factorize' is a Pollard-Rho factorization program.
|
||
|
||
* `isprime' is a command-line interface to the `mpz_probab_prime_p'
|
||
function.
|
||
|
||
* `primes' counts or lists primes in an interval, using a sieve.
|
||
|
||
* `qcn' is an example use of `mpz_kronecker_ui' to estimate quadratic
|
||
class numbers.
|
||
|
||
* The `perl' subdirectory is a comprehensive perl interface to MPIR.
|
||
See `demos/perl/INSTALL' for more information. Documentation is
|
||
in POD format in `demos/perl/GMP.pm'.
|
||
|
||
As an aside, consideration has been given at various times to some
|
||
sort of expression evaluation within the main MPIR library. Going
|
||
beyond something minimal quickly leads to matters like user-defined
|
||
functions, looping, fixnums for control variables, etc, which are
|
||
considered outside the scope of MPIR (much closer to language
|
||
interpreters or compilers, *Note Language Bindings::.) Something
|
||
simple for program input convenience may yet be a possibility, a
|
||
combination of the `expr' demo and the `pexpr' tree back-end perhaps.
|
||
But for now the above evaluators are offered as illustrations.
|
||
|
||
|
||
File: mpir.info, Node: Efficiency, Next: Debugging, Prev: Demonstration Programs, Up: MPIR Basics
|
||
|
||
3.11 Efficiency
|
||
===============
|
||
|
||
Small Operands
|
||
On small operands, the time for function call overheads and memory
|
||
allocation can be significant in comparison to actual calculation.
|
||
This is unavoidable in a general purpose variable precision
|
||
library, although MPIR attempts to be as efficient as it can on
|
||
both large and small operands.
|
||
|
||
Static Linking
|
||
On some CPUs, in particular the x86s, the static `libmpir.a'
|
||
should be used for maximum speed, since the PIC code in the shared
|
||
`libmpir.so' will have a small overhead on each function call and
|
||
global data address. For many programs this will be
|
||
insignificant, but for long calculations there's a gain to be had.
|
||
|
||
Initializing and Clearing
|
||
Avoid excessive initializing and clearing of variables, since this
|
||
can be quite time consuming, especially in comparison to otherwise
|
||
fast operations like addition.
|
||
|
||
A language interpreter might want to keep a free list or stack of
|
||
initialized variables ready for use. It should be possible to
|
||
integrate something like that with a garbage collector too.
|
||
|
||
Reallocations
|
||
An `mpz_t' or `mpq_t' variable used to hold successively increasing
|
||
values will have its memory repeatedly `realloc'ed, which could be
|
||
quite slow or could fragment memory, depending on the C library.
|
||
If an application can estimate the final size then `mpz_init2' or
|
||
`mpz_realloc2' can be called to allocate the necessary space from
|
||
the beginning (*note Initializing Integers::).
|
||
|
||
It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2'
|
||
is too small, since all functions will do a further reallocation
|
||
if necessary. Badly overestimating memory required will waste
|
||
space though.
|
||
|
||
`2exp' Functions
|
||
It's up to an application to call functions like `mpz_mul_2exp'
|
||
when appropriate. General purpose functions like `mpz_mul' make
|
||
no attempt to identify powers of two or other special forms,
|
||
because such inputs will usually be very rare and testing every
|
||
time would be wasteful.
|
||
|
||
`ui' and `si' Functions
|
||
The `ui' functions and the small number of `si' functions exist for
|
||
convenience and should be used where applicable. But if for
|
||
example an `mpz_t' contains a value that fits in an `unsigned
|
||
long' there's no need extract it and call a `ui' function, just
|
||
use the regular `mpz' function.
|
||
|
||
In-Place Operations
|
||
`mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and
|
||
`mpf_neg' are fast when used for in-place operations like
|
||
`mpz_abs(x,x)', since in the current implementation only a single
|
||
field of `x' needs changing. On suitable compilers (GCC for
|
||
instance) this is inlined too.
|
||
|
||
`mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit
|
||
from an in-place operation like `mpz_add_ui(x,x,y)', since usually
|
||
only one or two limbs of `x' will need to be changed. The same
|
||
applies to the full precision `mpz_add' etc if `y' is small. If
|
||
`y' is big then cache locality may be helped, but that's all.
|
||
|
||
`mpz_mul' is currently the opposite, a separate destination is
|
||
slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is
|
||
only one limb, make a temporary copy of `x' before forming the
|
||
result. Normally that copying will only be a tiny fraction of the
|
||
time for the multiply, so this is not a particularly important
|
||
consideration.
|
||
|
||
`mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no
|
||
attempt to recognise a copy of something to itself, so a call like
|
||
`mpz_set(x,x)' will be wasteful. Naturally that would never be
|
||
written deliberately, but if it might arise from two pointers to
|
||
the same object then a test to avoid it might be desirable.
|
||
|
||
if (x != y)
|
||
mpz_set (x, y);
|
||
|
||
Note that it's never worth introducing extra `mpz_set' calls just
|
||
to get in-place operations. If a result should go to a particular
|
||
variable then just direct it there and let MPIR take care of data
|
||
movement.
|
||
|
||
Divisibility Testing (Small Integers)
|
||
`mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best
|
||
functions for testing whether an `mpz_t' is divisible by an
|
||
individual small integer. They use an algorithm which is faster
|
||
than `mpz_tdiv_ui', but which gives no useful information about
|
||
the actual remainder, only whether it's zero (or a particular
|
||
value).
|
||
|
||
However when testing divisibility by several small integers, it's
|
||
best to take a remainder modulo their product, to save
|
||
multi-precision operations. For instance to test whether a number
|
||
is divisible by any of 23, 29 or 31 take a remainder modulo
|
||
23*29*31 = 20677 and then test that.
|
||
|
||
The division functions like `mpz_tdiv_q_ui' which give a quotient
|
||
as well as a remainder are generally a little slower than the
|
||
remainder-only functions like `mpz_tdiv_ui'. If the quotient is
|
||
only rarely wanted then it's probably best to just take a
|
||
remainder and then go back and calculate the quotient if and when
|
||
it's wanted (`mpz_divexact_ui' can be used if the remainder is
|
||
zero).
|
||
|
||
Rational Arithmetic
|
||
The `mpq' functions operate on `mpq_t' values with no common
|
||
factors in the numerator and denominator. Common factors are
|
||
checked-for and cast out as necessary. In general, cancelling
|
||
factors every time is the best approach since it minimizes the
|
||
sizes for subsequent operations.
|
||
|
||
However, applications that know something about the factorization
|
||
of the values they're working with might be able to avoid some of
|
||
the GCDs used for canonicalization, or swap them for divisions.
|
||
For example when multiplying by a prime it's enough to check for
|
||
factors of it in the denominator instead of doing a full GCD. Or
|
||
when forming a big product it might be known that very little
|
||
cancellation will be possible, and so canonicalization can be left
|
||
to the end.
|
||
|
||
The `mpq_numref' and `mpq_denref' macros give access to the
|
||
numerator and denominator to do things outside the scope of the
|
||
supplied `mpq' functions. *Note Applying Integer Functions::.
|
||
|
||
The canonical form for rationals allows mixed-type `mpq_t' and
|
||
integer additions or subtractions to be done directly with
|
||
multiples of the denominator. This will be somewhat faster than
|
||
`mpq_add'. For example,
|
||
|
||
/* mpq increment */
|
||
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
|
||
|
||
/* mpq += unsigned long */
|
||
mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
|
||
|
||
/* mpq -= mpz */
|
||
mpz_submul (mpq_numref(q), mpq_denref(q), z);
|
||
|
||
Number Sequences
|
||
Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
|
||
designed for calculating isolated values. If a range of values is
|
||
wanted it's probably best to call to get a starting point and
|
||
iterate from there.
|
||
|
||
Text Input/Output
|
||
Hexadecimal or octal are suggested for input or output in text
|
||
form. Power-of-2 bases like these can be converted much more
|
||
efficiently than other bases, like decimal. For big numbers
|
||
there's usually nothing of particular interest to be seen in the
|
||
digits, so the base doesn't matter much.
|
||
|
||
Maybe we can hope octal will one day become the normal base for
|
||
everyday use, as proposed by King Charles XII of Sweden and later
|
||
reformers.
|
||
|
||
|
||
File: mpir.info, Node: Debugging, Next: Profiling, Prev: Efficiency, Up: MPIR Basics
|
||
|
||
3.12 Debugging
|
||
==============
|
||
|
||
Stack Overflow
|
||
Depending on the system, a segmentation violation or bus error
|
||
might be the only indication of stack overflow. See
|
||
`--enable-alloca' choices in *note Build Options::, for how to
|
||
address this.
|
||
|
||
In new enough versions of GCC, `-fstack-check' may be able to
|
||
ensure an overflow is recognised by the system before too much
|
||
damage is done, or `-fstack-limit-symbol' or
|
||
`-fstack-limit-register' may be able to add checking if the system
|
||
itself doesn't do any (*note Options for Code Generation:
|
||
(gcc)Code Gen Options.). These options must be added to the
|
||
`CFLAGS' used in the MPIR build (*note Build Options::), adding
|
||
them just to an application will have no effect. Note also
|
||
they're a slowdown, adding overhead to each function call and each
|
||
stack allocation.
|
||
|
||
Heap Problems
|
||
The most likely cause of application problems with MPIR is heap
|
||
corruption. Failing to `init' MPIR variables will have
|
||
unpredictable effects, and corruption arising elsewhere in a
|
||
program may well affect MPIR. Initializing MPIR variables more
|
||
than once or failing to clear them will cause memory leaks.
|
||
|
||
In all such cases a `malloc' debugger is recommended. On a GNU or
|
||
BSD system the standard C library `malloc' has some diagnostic
|
||
facilities, see *note Allocation Debugging: (libc)Allocation
|
||
Debugging, or `man 3 malloc'. Other possibilities, in no
|
||
particular order, include
|
||
|
||
`http://www.inf.ethz.ch/personal/biere/projects/ccmalloc/'
|
||
`http://dmalloc.com/'
|
||
`http://www.perens.com/FreeSoftware/' (electric fence)
|
||
`http://packages.debian.org/fda'
|
||
`http://www.gnupdate.org/components/leakbug/'
|
||
`http://people.redhat.com/~otaylor/memprof/'
|
||
`http://www.cbmamiga.demon.co.uk/mpatrol/'
|
||
|
||
The MPIR default allocation routines in `memory.c' also have a
|
||
simple sentinel scheme which can be enabled with `#define DEBUG'
|
||
in that file. This is mainly designed for detecting buffer
|
||
overruns during MPIR development, but might find other uses.
|
||
|
||
Stack Backtraces
|
||
On some systems the compiler options MPIR uses by default can
|
||
interfere with debugging. In particular on x86 and 68k systems
|
||
`-fomit-frame-pointer' is used and this generally inhibits stack
|
||
backtracing. Recompiling without such options may help while
|
||
debugging, though the usual caveats about it potentially moving a
|
||
memory problem or hiding a compiler bug will apply.
|
||
|
||
GDB, the GNU Debugger
|
||
A sample `.gdbinit' is included in the distribution, showing how
|
||
to call some undocumented dump functions to print MPIR variables
|
||
from within GDB. Note that these functions shouldn't be used in
|
||
final application code since they're undocumented and may be
|
||
subject to incompatible changes in future versions of MPIR.
|
||
|
||
Source File Paths
|
||
MPIR has multiple source files with the same name, in different
|
||
directories. For example `mpz', `mpq' and `mpf' each have an
|
||
`init.c'. If the debugger can't already determine the right one
|
||
it may help to build with absolute paths on each C file. One way
|
||
to do that is to use a separate object directory with an absolute
|
||
path to the source directory.
|
||
|
||
cd /my/build/dir
|
||
/my/source/dir/gmp-1.0/configure
|
||
|
||
This works via `VPATH', and might require GNU `make'. Alternately
|
||
it might be possible to change the `.c.lo' rules appropriately.
|
||
|
||
Assertion Checking
|
||
The build option `--enable-assert' is available to add some
|
||
consistency checks to the library (see *note Build Options::).
|
||
These are likely to be of limited value to most applications.
|
||
Assertion failures are just as likely to indicate memory
|
||
corruption as a library or compiler bug.
|
||
|
||
Applications using the low-level `mpn' functions, however, will
|
||
benefit from `--enable-assert' since it adds checks on the
|
||
parameters of most such functions, many of which have subtle
|
||
restrictions on their usage. Note however that only the generic C
|
||
code has checks, not the assembler code, so CPU `none' should be
|
||
used for maximum checking.
|
||
|
||
Temporary Memory Checking
|
||
The build option `--enable-alloca=debug' arranges that each block
|
||
of temporary memory in MPIR is allocated with a separate call to
|
||
`malloc' (or the allocation function set with
|
||
`mp_set_memory_functions').
|
||
|
||
This can help a malloc debugger detect accesses outside the
|
||
intended bounds, or detect memory not released. In a normal
|
||
build, on the other hand, temporary memory is allocated in blocks
|
||
which MPIR divides up for its own use, or may be allocated with a
|
||
compiler builtin `alloca' which will go nowhere near any malloc
|
||
debugger hooks.
|
||
|
||
Maximum Debuggability
|
||
To summarize the above, an MPIR build for maximum debuggability
|
||
would be
|
||
|
||
./configure --disable-shared --enable-assert \
|
||
--enable-alloca=debug --host=none CFLAGS=-g
|
||
|
||
For C++, add `--enable-cxx CXXFLAGS=-g'.
|
||
|
||
Checker
|
||
The GCC checker (`http://savannah.gnu.org/projects/checker/') can
|
||
be used with MPIR. It contains a stub library which means MPIR
|
||
applications compiled with checker can use a normal MPIR build.
|
||
|
||
A build of MPIR with checking within MPIR itself can be made.
|
||
This will run very very slowly. On GNU/Linux for example,
|
||
|
||
./configure --host=none-pc-linux-gnu CC=checkergcc
|
||
|
||
`--host=none' must be used, since the MPIR assembler code doesn't
|
||
support the checking scheme. The MPIR C++ features cannot be
|
||
used, since current versions of checker (0.9.9.1) don't yet
|
||
support the standard C++ library.
|
||
|
||
Valgrind
|
||
The valgrind program (`http://valgrind.org/') is a memory checker
|
||
for x86s. It translates and emulates machine instructions to do
|
||
strong checks for uninitialized data (at the level of individual
|
||
bits), memory accesses through bad pointers, and memory leaks.
|
||
|
||
Recent versions of Valgrind are getting support for MMX and
|
||
SSE/SSE2 instructions, for past versions MPIR will need to be
|
||
configured not to use those, ie. for an x86 without them (for
|
||
instance plain `i486').
|
||
|
||
Other Problems
|
||
Any suspected bug in MPIR itself should be isolated to make sure
|
||
it's not an application problem, see *note Reporting Bugs::.
|
||
|
||
|
||
File: mpir.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: MPIR Basics
|
||
|
||
3.13 Profiling
|
||
==============
|
||
|
||
Running a program under a profiler is a good way to find where it's
|
||
spending most time and where improvements can be best sought. The
|
||
profiling choices for a MPIR build are as follows.
|
||
|
||
`--disable-profiling'
|
||
The default is to add nothing special for profiling.
|
||
|
||
It should be possible to just compile the mainline of a program
|
||
with `-p' and use `prof' to get a profile consisting of
|
||
timer-based sampling of the program counter. Most of the MPIR
|
||
assembler code has the necessary symbol information.
|
||
|
||
This approach has the advantage of minimizing interference with
|
||
normal program operation, but on most systems the resolution of
|
||
the sampling is quite low (10 milliseconds for instance),
|
||
requiring long runs to get accurate information.
|
||
|
||
`--enable-profiling=prof'
|
||
Build with support for the system `prof', which means `-p' added
|
||
to the `CFLAGS'.
|
||
|
||
This provides call counting in addition to program counter
|
||
sampling, which allows the most frequently called routines to be
|
||
identified, and an average time spent in each routine to be
|
||
determined.
|
||
|
||
The x86 assembler code has support for this option, but on other
|
||
processors the assembler routines will be as if compiled without
|
||
`-p' and therefore won't appear in the call counts.
|
||
|
||
On some systems, such as GNU/Linux, `-p' in fact means `-pg' and in
|
||
this case `--enable-profiling=gprof' described below should be used
|
||
instead.
|
||
|
||
`--enable-profiling=gprof'
|
||
Build with support for `gprof' (*note GNU gprof: (gprof)Top.),
|
||
which means `-pg' added to the `CFLAGS'.
|
||
|
||
This provides call graph construction in addition to call counting
|
||
and program counter sampling, which makes it possible to count
|
||
calls coming from different locations. For example the number of
|
||
calls to `mpn_mul' from `mpz_mul' versus the number from
|
||
`mpf_mul'. The program counter sampling is still flat though, so
|
||
only a total time in `mpn_mul' would be accumulated, not a
|
||
separate amount for each call site.
|
||
|
||
The x86 assembler code has support for this option, but on other
|
||
processors the assembler routines will be as if compiled without
|
||
`-pg' and therefore not be included in the call counts.
|
||
|
||
On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
|
||
incompatible, so the latter is omitted from the default flags in
|
||
that case, which might result in poorer code generation.
|
||
|
||
Incidentally, it should be possible to use the `gprof' program
|
||
with a plain `--enable-profiling=prof' build. But in that case
|
||
only the `gprof -p' flat profile and call counts can be expected
|
||
to be valid, not the `gprof -q' call graph.
|
||
|
||
`--enable-profiling=instrument'
|
||
Build with the GCC option `-finstrument-functions' added to the
|
||
`CFLAGS' (*note Options for Code Generation: (gcc)Code Gen
|
||
Options.).
|
||
|
||
This inserts special instrumenting calls at the start and end of
|
||
each function, allowing exact timing and full call graph
|
||
construction.
|
||
|
||
This instrumenting is not normally a standard system feature and
|
||
will require support from an external library, such as
|
||
|
||
`http://sourceforge.net/projects/fnccheck/'
|
||
|
||
This should be included in `LIBS' during the MPIR configure so
|
||
that test programs will link. For example,
|
||
|
||
./configure --enable-profiling=instrument LIBS=-lfc
|
||
|
||
On a GNU system the C library provides dummy instrumenting
|
||
functions, so programs compiled with this option will link. In
|
||
this case it's only necessary to ensure the correct library is
|
||
added when linking an application.
|
||
|
||
The x86 assembler code supports this option, but on other
|
||
processors the assembler routines will be as if compiled without
|
||
`-finstrument-functions' meaning time spent in them will
|
||
effectively be attributed to their caller.
|
||
|
||
|
||
File: mpir.info, Node: Autoconf, Next: Emacs, Prev: Profiling, Up: MPIR Basics
|
||
|
||
3.14 Autoconf
|
||
=============
|
||
|
||
Autoconf based applications can easily check whether MPIR is installed.
|
||
The only thing to be noted is that GMP/MPIR library symbols from
|
||
version 3 of GMP and version 1 of MPIR onwards have prefixes like
|
||
`__gmpz'. The following therefore would be a simple test,
|
||
|
||
AC_CHECK_LIB(mpir, __gmpz_init)
|
||
|
||
This just uses the default `AC_CHECK_LIB' actions for found or not
|
||
found, but an application that must have MPIR would want to generate an
|
||
error if not found. For example,
|
||
|
||
AC_CHECK_LIB(mpir, __gmpz_init, ,
|
||
[AC_MSG_ERROR([MPIR not found, see http://www.mpir.org/])])
|
||
|
||
If functions added in some particular version of GMP/MPIR are
|
||
required, then one of those can be used when checking. For example
|
||
`mpz_mul_si' was added in GMP 3.1,
|
||
|
||
AC_CHECK_LIB(mpir, __gmpz_mul_si, ,
|
||
[AC_MSG_ERROR(
|
||
[GMP/MPIR not found, or not GMP 3.1 or up or MPIR 1.0 or up, see http://www.mpir.org/])])
|
||
|
||
An alternative would be to test the version number in `mpir.h' using
|
||
say `AC_EGREP_CPP'. That would make it possible to test the exact
|
||
version, if some particular sub-minor release is known to be necessary.
|
||
|
||
In general it's recommended that applications should simply demand a
|
||
new enough MPIR rather than trying to provide supplements for features
|
||
not available in past versions.
|
||
|
||
Occasionally an application will need or want to know the size of a
|
||
type at configuration or preprocessing time, not just with `sizeof' in
|
||
the code. This can be done in the normal way with `mp_limb_t' etc, but
|
||
GMP 4.0 or up and MPIR 1.0 and up is best for this, since prior
|
||
versions needed certain `-D' defines on systems using a `long long'
|
||
limb. The following would suit Autoconf 2.50 or up,
|
||
|
||
AC_CHECK_SIZEOF(mp_limb_t, , [#include <mpir.h>])
|
||
|
||
|
||
File: mpir.info, Node: Emacs, Prev: Autoconf, Up: MPIR Basics
|
||
|
||
3.15 Emacs
|
||
==========
|
||
|
||
<C-h C-i> (`info-lookup-symbol') is a good way to find documentation on
|
||
C functions while editing (*note Info Documentation Lookup: (emacs)Info
|
||
Lookup.).
|
||
|
||
The MPIR manual can be included in such lookups by putting the
|
||
following in your `.emacs',
|
||
|
||
(eval-after-load "info-look"
|
||
'(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
|
||
(setcar (nthcdr 3 mode-value)
|
||
(cons '("(gmp)Function Index" nil "^ -.* " "\\>")
|
||
(nth 3 mode-value)))))
|
||
|
||
|
||
File: mpir.info, Node: Reporting Bugs, Next: Integer Functions, Prev: MPIR Basics, Up: Top
|
||
|
||
4 Reporting Bugs
|
||
****************
|
||
|
||
If you think you have found a bug in the MPIR library, please
|
||
investigate it and report it. We have made this library available to
|
||
you, and it is not too much to ask you to report the bugs you find.
|
||
|
||
Before you report a bug, check it's not already addressed in *note
|
||
Known Build Problems::, or perhaps *note Notes for Particular
|
||
Systems::. You may also want to check `http://www.mpir.org/' for
|
||
patches for this release.
|
||
|
||
Please include the following in any report,
|
||
|
||
* The MPIR version number, and if pre-packaged or patched then say
|
||
so.
|
||
|
||
* A test program that makes it possible for us to reproduce the bug.
|
||
Include instructions on how to run the program.
|
||
|
||
* A description of what is wrong. If the results are incorrect, in
|
||
what way. If you get a crash, say so.
|
||
|
||
* If you get a crash, include a stack backtrace from the debugger if
|
||
it's informative (`where' in `gdb', or `$C' in `adb').
|
||
|
||
* Please do not send core dumps, executables or `strace's.
|
||
|
||
* The configuration options you used when building MPIR, if any.
|
||
|
||
* The name of the compiler and its version. For `gcc', get the
|
||
version with `gcc -v', otherwise perhaps `what `which cc`', or
|
||
similar.
|
||
|
||
* The output from running `uname -a'.
|
||
|
||
* The output from running `./config.guess', and from running
|
||
`./configfsf.guess' (might be the same).
|
||
|
||
* If the bug is related to `configure', then the contents of
|
||
`config.log'.
|
||
|
||
* If the bug is related to an `asm' file not assembling, then the
|
||
contents of `config.m4' and the offending line or lines from the
|
||
temporary `mpn/tmp-<file>.s'.
|
||
|
||
Please make an effort to produce a self-contained report, with
|
||
something definite that can be tested or debugged. Vague queries or
|
||
piecemeal messages are difficult to act on and don't help the
|
||
development effort.
|
||
|
||
It is not uncommon that an observed problem is actually due to a bug
|
||
in the compiler; the MPIR code tends to explore interesting corners in
|
||
compilers.
|
||
|
||
If your bug report is good, we will do our best to help you get a
|
||
corrected version of the library; if the bug report is poor, we won't
|
||
do anything about it (except maybe ask you to send a better report).
|
||
|
||
Send your report to: `http://groups.google.com/group/mpir-devel'.
|
||
|
||
If you think something in this manual is unclear, or downright
|
||
incorrect, or if the language needs to be improved, please send a note
|
||
to the same address.
|
||
|
||
|
||
File: mpir.info, Node: Integer Functions, Next: Rational Number Functions, Prev: Reporting Bugs, Up: Top
|
||
|
||
5 Integer Functions
|
||
*******************
|
||
|
||
This chapter describes the MPIR functions for performing integer
|
||
arithmetic. These functions start with the prefix `mpz_'.
|
||
|
||
MPIR integers are stored in objects of type `mpz_t'.
|
||
|
||
* Menu:
|
||
|
||
* Initializing Integers::
|
||
* Assigning Integers::
|
||
* Simultaneous Integer Init & Assign::
|
||
* Converting Integers::
|
||
* Integer Arithmetic::
|
||
* Integer Division::
|
||
* Integer Exponentiation::
|
||
* Integer Roots::
|
||
* Number Theoretic Functions::
|
||
* Integer Comparisons::
|
||
* Integer Logic and Bit Fiddling::
|
||
* I/O of Integers::
|
||
* Integer Random Numbers::
|
||
* Integer Import and Export::
|
||
* Miscellaneous Integer Functions::
|
||
* Integer Special Functions::
|
||
|
||
|
||
File: mpir.info, Node: Initializing Integers, Next: Assigning Integers, Prev: Integer Functions, Up: Integer Functions
|
||
|
||
5.1 Initialization Functions
|
||
============================
|
||
|
||
The functions for integer arithmetic assume that all integer objects are
|
||
initialized. You do that by calling the function `mpz_init'. For
|
||
example,
|
||
|
||
{
|
||
mpz_t integ;
|
||
mpz_init (integ);
|
||
...
|
||
mpz_add (integ, ...);
|
||
...
|
||
mpz_sub (integ, ...);
|
||
|
||
/* Unless the program is about to exit, do ... */
|
||
mpz_clear (integ);
|
||
}
|
||
|
||
As you can see, you can store new values any number of times, once an
|
||
object is initialized.
|
||
|
||
-- Function: void mpz_init (mpz_t INTEGER)
|
||
Initialize INTEGER, and set its value to 0.
|
||
|
||
-- Function: void mpz_init2 (mpz_t INTEGER, unsigned long N)
|
||
Initialize INTEGER, with space for N bits, and set its value to 0.
|
||
|
||
N is only the initial space, INTEGER will grow automatically in
|
||
the normal way, if necessary, for subsequent values stored.
|
||
`mpz_init2' makes it possible to avoid such reallocations if a
|
||
maximum size is known in advance.
|
||
|
||
-- Function: void mpz_clear (mpz_t INTEGER)
|
||
Free the space occupied by INTEGER. Call this function for all
|
||
`mpz_t' variables when you are done with them.
|
||
|
||
-- Function: void mpz_realloc2 (mpz_t INTEGER, unsigned long N)
|
||
Change the space allocated for INTEGER to N bits. The value in
|
||
INTEGER is preserved if it fits, or is set to 0 if not.
|
||
|
||
This function can be used to increase the space for a variable in
|
||
order to avoid repeated automatic reallocations, or to decrease it
|
||
to give memory back to the heap.
|
||
|
||
|
||
File: mpir.info, Node: Assigning Integers, Next: Simultaneous Integer Init & Assign, Prev: Initializing Integers, Up: Integer Functions
|
||
|
||
5.2 Assignment Functions
|
||
========================
|
||
|
||
These functions assign new values to already initialized integers
|
||
(*note Initializing Integers::).
|
||
|
||
-- Function: void mpz_set (mpz_t ROP, mpz_t OP)
|
||
-- Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP)
|
||
-- Function: void mpz_set_si (mpz_t ROP, signed long int OP)
|
||
-- Function: void mpz_set_d (mpz_t ROP, double OP)
|
||
-- Function: void mpz_set_q (mpz_t ROP, mpq_t OP)
|
||
-- Function: void mpz_set_f (mpz_t ROP, mpf_t OP)
|
||
Set the value of ROP from OP.
|
||
|
||
`mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an
|
||
integer.
|
||
|
||
-- Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE)
|
||
Set the value of ROP from STR, a null-terminated C string in base
|
||
BASE. White space is allowed in the string, and is simply ignored.
|
||
|
||
The BASE may vary from 2 to 62, or if BASE is 0, then the leading
|
||
characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
|
||
for binary, `0' for octal, or decimal otherwise.
|
||
|
||
For bases up to 36, case is ignored; upper-case and lower-case
|
||
letters have the same value. For bases 37 to 62, upper-case
|
||
letter represent the usual 10..35 while lower-case letter
|
||
represent 36..61.
|
||
|
||
This function returns 0 if the entire string is a valid number in
|
||
base BASE. Otherwise it returns -1.
|
||
|
||
-- Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2)
|
||
Swap the values ROP1 and ROP2 efficiently.
|
||
|
||
|
||
File: mpir.info, Node: Simultaneous Integer Init & Assign, Next: Converting Integers, Prev: Assigning Integers, Up: Integer Functions
|
||
|
||
5.3 Combined Initialization and Assignment Functions
|
||
====================================================
|
||
|
||
For convenience, MPIR provides a parallel series of initialize-and-set
|
||
functions which initialize the output and then store the value there.
|
||
These functions' names have the form `mpz_init_set...'
|
||
|
||
Here is an example of using one:
|
||
|
||
{
|
||
mpz_t pie;
|
||
mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
|
||
...
|
||
mpz_sub (pie, ...);
|
||
...
|
||
mpz_clear (pie);
|
||
}
|
||
|
||
Once the integer has been initialized by any of the `mpz_init_set...'
|
||
functions, it can be used as the source or destination operand for the
|
||
ordinary integer functions. Don't use an initialize-and-set function
|
||
on a variable already initialized!
|
||
|
||
-- Function: void mpz_init_set (mpz_t ROP, mpz_t OP)
|
||
-- Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP)
|
||
-- Function: void mpz_init_set_si (mpz_t ROP, signed long int OP)
|
||
-- Function: void mpz_init_set_d (mpz_t ROP, double OP)
|
||
Initialize ROP with limb space and set the initial numeric value
|
||
from OP.
|
||
|
||
-- Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE)
|
||
Initialize ROP and set its value like `mpz_set_str' (see its
|
||
documentation above for details).
|
||
|
||
If the string is a correct base BASE number, the function returns
|
||
0; if an error occurs it returns -1. ROP is initialized even if
|
||
an error occurs. (I.e., you have to call `mpz_clear' for it.)
|
||
|
||
|
||
File: mpir.info, Node: Converting Integers, Next: Integer Arithmetic, Prev: Simultaneous Integer Init & Assign, Up: Integer Functions
|
||
|
||
5.4 Conversion Functions
|
||
========================
|
||
|
||
This section describes functions for converting MPIR integers to
|
||
standard C types. Functions for converting _to_ MPIR integers are
|
||
described in *note Assigning Integers:: and *note I/O of Integers::.
|
||
|
||
-- Function: unsigned long int mpz_get_ui (mpz_t OP)
|
||
Return the value of OP as an `unsigned long'.
|
||
|
||
If OP is too big to fit an `unsigned long' then just the least
|
||
significant bits that do fit are returned. The sign of OP is
|
||
ignored, only the absolute value is used.
|
||
|
||
-- Function: signed long int mpz_get_si (mpz_t OP)
|
||
If OP fits into a `signed long int' return the value of OP.
|
||
Otherwise return the least significant part of OP, with the same
|
||
sign as OP.
|
||
|
||
If OP is too big to fit in a `signed long int', the returned
|
||
result is probably not very useful. To find out if the value will
|
||
fit, use the function `mpz_fits_slong_p'.
|
||
|
||
-- Function: double mpz_get_d (mpz_t OP)
|
||
Convert OP to a `double', truncating if necessary (ie. rounding
|
||
towards zero).
|
||
|
||
If the exponent from the conversion is too big, the result is
|
||
system dependent. An infinity is returned where available. A
|
||
hardware overflow trap may or may not occur.
|
||
|
||
-- Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP)
|
||
Convert OP to a `double', truncating if necessary (ie. rounding
|
||
towards zero), and returning the exponent separately.
|
||
|
||
The return value is in the range 0.5<=abs(D)<1 and the exponent is
|
||
stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
|
||
is zero, the return is 0.0 and 0 is stored to `*EXP'.
|
||
|
||
This is similar to the standard C `frexp' function (*note
|
||
Normalization Functions: (libc)Normalization Functions.).
|
||
|
||
-- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
|
||
Convert OP to a string of digits in base BASE. The base may vary
|
||
from 2 to 36.
|
||
|
||
If STR is `NULL', the result string is allocated using the current
|
||
allocation function (*note Custom Allocation::). The block will be
|
||
`strlen(str)+1' bytes, that being exactly enough for the string and
|
||
null-terminator.
|
||
|
||
If STR is not `NULL', it should point to a block of storage large
|
||
enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'.
|
||
The two extra bytes are for a possible minus sign, and the
|
||
null-terminator.
|
||
|
||
A pointer to the result string is returned, being either the
|
||
allocated block, or the given STR.
|
||
|
||
|
||
File: mpir.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions
|
||
|
||
5.5 Arithmetic Functions
|
||
========================
|
||
|
||
-- Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
-- Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int
|
||
OP2)
|
||
Set ROP to OP1 + OP2.
|
||
|
||
-- Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
-- Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int
|
||
OP2)
|
||
-- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t
|
||
OP2)
|
||
Set ROP to OP1 - OP2.
|
||
|
||
-- Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
-- Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2)
|
||
-- Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int
|
||
OP2)
|
||
Set ROP to OP1 times OP2.
|
||
|
||
-- Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
-- Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long
|
||
int OP2)
|
||
Set ROP to ROP + OP1 times OP2.
|
||
|
||
-- Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
-- Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long
|
||
int OP2)
|
||
Set ROP to ROP - OP1 times OP2.
|
||
|
||
-- Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, unsigned long
|
||
int OP2)
|
||
Set ROP to OP1 times 2 raised to OP2. This operation can also be
|
||
defined as a left shift by OP2 bits.
|
||
|
||
-- Function: void mpz_neg (mpz_t ROP, mpz_t OP)
|
||
Set ROP to -OP.
|
||
|
||
-- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
|
||
Set ROP to the absolute value of OP.
|
||
|
||
|
||
File: mpir.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions
|
||
|
||
5.6 Division Functions
|
||
======================
|
||
|
||
Division is undefined if the divisor is zero. Passing a zero divisor
|
||
to the division or modulo functions (including the modular powering
|
||
functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional
|
||
division by zero. This lets a program handle arithmetic exceptions in
|
||
these functions the same way as for normal C `int' arithmetic.
|
||
|
||
-- Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D)
|
||
-- Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D)
|
||
-- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
|
||
-- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
|
||
mpz_t N, unsigned long int D)
|
||
-- Function: unsigned long int mpz_cdiv_ui (mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N,
|
||
unsigned long int B)
|
||
-- Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N,
|
||
unsigned long int B)
|
||
|
||
-- Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D)
|
||
-- Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D)
|
||
-- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
|
||
-- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
|
||
mpz_t N, unsigned long int D)
|
||
-- Function: unsigned long int mpz_fdiv_ui (mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N,
|
||
unsigned long int B)
|
||
-- Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N,
|
||
unsigned long int B)
|
||
|
||
-- Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D)
|
||
-- Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D)
|
||
-- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
|
||
-- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
|
||
mpz_t N, unsigned long int D)
|
||
-- Function: unsigned long int mpz_tdiv_ui (mpz_t N,
|
||
unsigned long int D)
|
||
-- Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N,
|
||
unsigned long int B)
|
||
-- Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N,
|
||
unsigned long int B)
|
||
|
||
Divide N by D, forming a quotient Q and/or remainder R. For the
|
||
`2exp' functions, D=2^B. The rounding is in three styles, each
|
||
suiting different applications.
|
||
|
||
* `cdiv' rounds Q up towards +infinity, and R will have the
|
||
opposite sign to D. The `c' stands for "ceil".
|
||
|
||
* `fdiv' rounds Q down towards -infinity, and R will have the
|
||
same sign as D. The `f' stands for "floor".
|
||
|
||
* `tdiv' rounds Q towards zero, and R will have the same sign
|
||
as N. The `t' stands for "truncate".
|
||
|
||
In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
|
||
0<=abs(R)<abs(D).
|
||
|
||
The `q' functions calculate only the quotient, the `r' functions
|
||
only the remainder, and the `qr' functions calculate both. Note
|
||
that for `qr' the same variable cannot be passed for both Q and R,
|
||
or results will be unpredictable.
|
||
|
||
For the `ui' variants the return value is the remainder, and in
|
||
fact returning the remainder is all the `div_ui' functions do. For
|
||
`tdiv' and `cdiv' the remainder can be negative, so for those the
|
||
return value is the absolute value of the remainder.
|
||
|
||
For the `2exp' variants the divisor is 2^B. These functions are
|
||
implemented as right shifts and bit masks, but of course they
|
||
round the same as the other functions.
|
||
|
||
For positive N both `mpz_fdiv_q_2exp' and `mpz_tdiv_q_2exp' are
|
||
simple bitwise right shifts. For negative N, `mpz_fdiv_q_2exp' is
|
||
effectively an arithmetic right shift treating N as twos complement
|
||
the same as the bitwise logical functions do, whereas
|
||
`mpz_tdiv_q_2exp' effectively treats N as sign and magnitude.
|
||
|
||
-- Function: void mpz_mod (mpz_t R, mpz_t N, mpz_t D)
|
||
-- Function: unsigned long int mpz_mod_ui (mpz_t R, mpz_t N,
|
||
unsigned long int D)
|
||
Set R to N `mod' D. The sign of the divisor is ignored; the
|
||
result is always non-negative.
|
||
|
||
`mpz_mod_ui' is identical to `mpz_fdiv_r_ui' above, returning the
|
||
remainder as well as setting R. See `mpz_fdiv_ui' above if only
|
||
the return value is wanted.
|
||
|
||
-- Function: void mpz_divexact (mpz_t Q, mpz_t N, mpz_t D)
|
||
-- Function: void mpz_divexact_ui (mpz_t Q, mpz_t N, unsigned long D)
|
||
Set Q to N/D. These functions produce correct results only when
|
||
it is known in advance that D divides N.
|
||
|
||
These routines are much faster than the other division functions,
|
||
and are the best choice when exact division is known to occur, for
|
||
example reducing a rational to lowest terms.
|
||
|
||
-- Function: int mpz_divisible_p (mpz_t N, mpz_t D)
|
||
-- Function: int mpz_divisible_ui_p (mpz_t N, unsigned long int D)
|
||
-- Function: int mpz_divisible_2exp_p (mpz_t N, unsigned long int B)
|
||
Return non-zero if N is exactly divisible by D, or in the case of
|
||
`mpz_divisible_2exp_p' by 2^B.
|
||
|
||
N is divisible by D if there exists an integer Q satisfying N =
|
||
Q*D. Unlike the other division functions, D=0 is accepted and
|
||
following the rule it can be seen that only 0 is considered
|
||
divisible by 0.
|
||
|
||
-- Function: int mpz_congruent_p (mpz_t N, mpz_t C, mpz_t D)
|
||
-- Function: int mpz_congruent_ui_p (mpz_t N, unsigned long int C,
|
||
unsigned long int D)
|
||
-- Function: int mpz_congruent_2exp_p (mpz_t N, mpz_t C, unsigned long
|
||
int B)
|
||
Return non-zero if N is congruent to C modulo D, or in the case of
|
||
`mpz_congruent_2exp_p' modulo 2^B.
|
||
|
||
N is congruent to C mod D if there exists an integer Q satisfying
|
||
N = C + Q*D. Unlike the other division functions, D=0 is accepted
|
||
and following the rule it can be seen that N and C are considered
|
||
congruent mod 0 only when exactly equal.
|
||
|
||
|
||
File: mpir.info, Node: Integer Exponentiation, Next: Integer Roots, Prev: Integer Division, Up: Integer Functions
|
||
|
||
5.7 Exponentiation Functions
|
||
============================
|
||
|
||
-- Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t
|
||
MOD)
|
||
-- Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long
|
||
int EXP, mpz_t MOD)
|
||
Set ROP to (BASE raised to EXP) modulo MOD.
|
||
|
||
Negative EXP is supported if an inverse BASE^-1 mod MOD exists
|
||
(see `mpz_invert' in *note Number Theoretic Functions::). If an
|
||
inverse doesn't exist then a divide by zero is raised.
|
||
|
||
-- Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int
|
||
EXP)
|
||
-- Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
|
||
unsigned long int EXP)
|
||
Set ROP to BASE raised to EXP. The case 0^0 yields 1.
|
||
|
||
|
||
File: mpir.info, Node: Integer Roots, Next: Number Theoretic Functions, Prev: Integer Exponentiation, Up: Integer Functions
|
||
|
||
5.8 Root Extraction Functions
|
||
=============================
|
||
|
||
-- Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N)
|
||
Set ROP to the truncated integer part of the Nth root of OP.
|
||
Return non-zero if the computation was exact, i.e., if OP is ROP
|
||
to the Nth power.
|
||
|
||
-- Function: void mpz_rootrem (mpz_t ROOT, mpz_t REM, mpz_t U,
|
||
unsigned long int N)
|
||
Set ROOT to the truncated integer part of the Nth root of U. Set
|
||
REM to the remainder, U-ROOT**N.
|
||
|
||
-- Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)
|
||
Set ROP to the truncated integer part of the square root of OP.
|
||
|
||
-- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)
|
||
Set ROP1 to the truncated integer part of the square root of OP,
|
||
like `mpz_sqrt'. Set ROP2 to the remainder OP-ROP1*ROP1, which
|
||
will be zero if OP is a perfect square.
|
||
|
||
If ROP1 and ROP2 are the same variable, the results are undefined.
|
||
|
||
-- Function: int mpz_perfect_power_p (mpz_t OP)
|
||
Return non-zero if OP is a perfect power, i.e., if there exist
|
||
integers A and B, with B>1, such that OP equals A raised to the
|
||
power B.
|
||
|
||
Under this definition both 0 and 1 are considered to be perfect
|
||
powers. Negative values of OP are accepted, but of course can
|
||
only be odd perfect powers.
|
||
|
||
-- Function: int mpz_perfect_square_p (mpz_t OP)
|
||
Return non-zero if OP is a perfect square, i.e., if the square
|
||
root of OP is an integer. Under this definition both 0 and 1 are
|
||
considered to be perfect squares.
|
||
|
||
|
||
File: mpir.info, Node: Number Theoretic Functions, Next: Integer Comparisons, Prev: Integer Roots, Up: Integer Functions
|
||
|
||
5.9 Number Theoretic Functions
|
||
==============================
|
||
|
||
-- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
|
||
Determine whether N is prime. Return 2 if N is definitely prime,
|
||
return 1 if N is probably prime (without being certain), or return
|
||
0 if N is definitely composite.
|
||
|
||
This function does some trial divisions, then some Miller-Rabin
|
||
probabilistic primality tests. REPS controls how many such tests
|
||
are done, 5 to 10 is a reasonable number, more will reduce the
|
||
chances of a composite being returned as "probably prime".
|
||
|
||
Miller-Rabin and similar tests can be more properly called
|
||
compositeness tests. Numbers which fail are known to be composite
|
||
but those which pass might be prime or might be composite. Only a
|
||
few composites pass, hence those which pass are considered
|
||
probably prime.
|
||
|
||
-- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
|
||
Set ROP to the next prime greater than OP.
|
||
|
||
This function uses a probabilistic algorithm to identify primes.
|
||
For practical purposes it's adequate, the chance of a composite
|
||
passing will be extremely small.
|
||
|
||
-- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
Set ROP to the greatest common divisor of OP1 and OP2. The result
|
||
is always positive even if one or both input operands are negative.
|
||
|
||
-- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
|
||
unsigned long int OP2)
|
||
Compute the greatest common divisor of OP1 and OP2. If ROP is not
|
||
`NULL', store the result there.
|
||
|
||
If the result is small enough to fit in an `unsigned long int', it
|
||
is returned. If the result does not fit, 0 is returned, and the
|
||
result is equal to the argument OP1. Note that the result will
|
||
always fit if OP2 is non-zero.
|
||
|
||
-- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A,
|
||
mpz_t B)
|
||
Set G to the greatest common divisor of A and B, and in addition
|
||
set S and T to coefficients satisfying A*S + B*T = G. G is always
|
||
positive, even if one or both of A and B are negative.
|
||
|
||
If T is `NULL' then that value is not computed.
|
||
|
||
-- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
-- Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2)
|
||
Set ROP to the least common multiple of OP1 and OP2. ROP is
|
||
always positive, irrespective of the signs of OP1 and OP2. ROP
|
||
will be zero if either OP1 or OP2 is zero.
|
||
|
||
-- Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
Compute the inverse of OP1 modulo OP2 and put the result in ROP.
|
||
If the inverse exists, the return value is non-zero and ROP will
|
||
satisfy 0 <= ROP < OP2. If an inverse doesn't exist the return
|
||
value is zero and ROP is undefined.
|
||
|
||
-- Function: int mpz_jacobi (mpz_t A, mpz_t B)
|
||
Calculate the Jacobi symbol (A/B). This is defined only for B odd.
|
||
|
||
-- Function: int mpz_legendre (mpz_t A, mpz_t P)
|
||
Calculate the Legendre symbol (A/P). This is defined only for P
|
||
an odd positive prime, and for such P it's identical to the Jacobi
|
||
symbol.
|
||
|
||
-- Function: int mpz_kronecker (mpz_t A, mpz_t B)
|
||
-- Function: int mpz_kronecker_si (mpz_t A, long B)
|
||
-- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B)
|
||
-- Function: int mpz_si_kronecker (long A, mpz_t B)
|
||
-- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B)
|
||
Calculate the Jacobi symbol (A/B) with the Kronecker extension
|
||
(a/2)=(2/a) when a odd, or (a/2)=0 when a even.
|
||
|
||
When B is odd the Jacobi symbol and Kronecker symbol are
|
||
identical, so `mpz_kronecker_ui' etc can be used for mixed
|
||
precision Jacobi symbols too.
|
||
|
||
For more information see Henri Cohen section 1.4.2 (*note
|
||
References::), or any number theory textbook. See also the
|
||
example program `demos/qcn.c' which uses `mpz_kronecker_ui'.
|
||
|
||
-- Function: unsigned long int mpz_remove (mpz_t ROP, mpz_t OP, mpz_t
|
||
F)
|
||
Remove all occurrences of the factor F from OP and store the
|
||
result in ROP. The return value is how many such occurrences were
|
||
removed.
|
||
|
||
-- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
|
||
Set ROP to OP!, the factorial of OP.
|
||
|
||
-- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
|
||
-- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
|
||
unsigned long int K)
|
||
Compute the binomial coefficient N over K and store the result in
|
||
ROP. Negative values of N are supported by `mpz_bin_ui', using
|
||
the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1
|
||
section 1.2.6 part G.
|
||
|
||
-- Function: void mpz_fib_ui (mpz_t FN, unsigned long int N)
|
||
-- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
|
||
int N)
|
||
`mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
|
||
`mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
|
||
|
||
These functions are designed for calculating isolated Fibonacci
|
||
numbers. When a sequence of values is wanted it's best to start
|
||
with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
|
||
similar.
|
||
|
||
-- Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N)
|
||
-- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned
|
||
long int N)
|
||
`mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
|
||
`mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
|
||
|
||
These functions are designed for calculating isolated Lucas
|
||
numbers. When a sequence of values is wanted it's best to start
|
||
with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
|
||
or similar.
|
||
|
||
The Fibonacci numbers and Lucas numbers are related sequences, so
|
||
it's never necessary to call both `mpz_fib2_ui' and
|
||
`mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
|
||
can be found in *note Lucas Numbers Algorithm::, the reverse is
|
||
straightforward too.
|
||
|
||
|
||
File: mpir.info, Node: Integer Comparisons, Next: Integer Logic and Bit Fiddling, Prev: Number Theoretic Functions, Up: Integer Functions
|
||
|
||
5.10 Comparison Functions
|
||
=========================
|
||
|
||
-- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
|
||
-- Function: int mpz_cmp_d (mpz_t OP1, double OP2)
|
||
-- Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)
|
||
-- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
|
||
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
|
||
if OP1 = OP2, or a negative value if OP1 < OP2.
|
||
|
||
`mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their
|
||
arguments more than once. `mpz_cmp_d' can be called with an
|
||
infinity, but results are undefined for a NaN.
|
||
|
||
-- Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
|
||
-- Function: int mpz_cmpabs_d (mpz_t OP1, double OP2)
|
||
-- Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)
|
||
Compare the absolute values of OP1 and OP2. Return a positive
|
||
value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a
|
||
negative value if abs(OP1) < abs(OP2).
|
||
|
||
`mpz_cmpabs_d' can be called with an infinity, but results are
|
||
undefined for a NaN.
|
||
|
||
-- Macro: int mpz_sgn (mpz_t OP)
|
||
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
|
||
|
||
This function is actually implemented as a macro. It evaluates
|
||
its argument multiple times.
|
||
|
||
|
||
File: mpir.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
|
||
|
||
5.11 Logical and Bit Manipulation Functions
|
||
===========================================
|
||
|
||
These functions behave as if twos complement arithmetic were used
|
||
(although sign-magnitude is the actual implementation). The least
|
||
significant bit is number 0.
|
||
|
||
-- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
Set ROP to OP1 bitwise-and OP2.
|
||
|
||
-- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
Set ROP to OP1 bitwise inclusive-or OP2.
|
||
|
||
-- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
|
||
Set ROP to OP1 bitwise exclusive-or OP2.
|
||
|
||
-- Function: void mpz_com (mpz_t ROP, mpz_t OP)
|
||
Set ROP to the one's complement of OP.
|
||
|
||
-- Function: unsigned long int mpz_popcount (mpz_t OP)
|
||
If OP>=0, return the population count of OP, which is the number
|
||
of 1 bits in the binary representation. If OP<0, the number of 1s
|
||
is infinite, and the return value is ULONG_MAX, the largest
|
||
possible `unsigned long'.
|
||
|
||
-- Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2)
|
||
If OP1 and OP2 are both >=0 or both <0, return the hamming
|
||
distance between the two operands, which is the number of bit
|
||
positions where OP1 and OP2 have different bit values. If one
|
||
operand is >=0 and the other <0 then the number of bits different
|
||
is infinite, and the return value is ULONG_MAX, the largest
|
||
possible `unsigned long'.
|
||
|
||
-- Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int
|
||
STARTING_BIT)
|
||
-- Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int
|
||
STARTING_BIT)
|
||
Scan OP, starting from bit STARTING_BIT, towards more significant
|
||
bits, until the first 0 or 1 bit (respectively) is found. Return
|
||
the index of the found bit.
|
||
|
||
If the bit at STARTING_BIT is already what's sought, then
|
||
STARTING_BIT is returned.
|
||
|
||
If there's no bit found, then ULONG_MAX is returned. This will
|
||
happen in `mpz_scan0' past the end of a positive number, or
|
||
`mpz_scan1' past the end of a negative.
|
||
|
||
-- Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX)
|
||
Set bit BIT_INDEX in ROP.
|
||
|
||
-- Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX)
|
||
Clear bit BIT_INDEX in ROP.
|
||
|
||
-- Function: void mpz_combit (mpz_t ROP, unsigned long int BIT_INDEX)
|
||
Complement bit BIT_INDEX in ROP.
|
||
|
||
-- Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX)
|
||
Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
|
||
|
||
|
||
File: mpir.info, Node: I/O of Integers, Next: Integer Random Numbers, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions
|
||
|
||
5.12 Input and Output Functions
|
||
===============================
|
||
|
||
Functions that perform input from a stdio stream, and functions that
|
||
output to a stdio stream. Passing a `NULL' pointer for a STREAM
|
||
argument to any of these functions will make them read from `stdin' and
|
||
write to `stdout', respectively.
|
||
|
||
When using any of these functions, it is a good idea to include
|
||
`stdio.h' before `mpir.h', since that will allow `mpir.h' to define
|
||
prototypes for these functions.
|
||
|
||
-- Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
|
||
Output OP on stdio stream STREAM, as a string of digits in base
|
||
BASE. The base may vary from 2 to 36.
|
||
|
||
Return the number of bytes written, or if an error occurred,
|
||
return 0.
|
||
|
||
-- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
|
||
Input a possibly white-space preceded string in base BASE from
|
||
stdio stream STREAM, and put the read integer in ROP.
|
||
|
||
The BASE may vary from 2 to 62, or if BASE is 0, then the leading
|
||
characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
|
||
for binary, `0' for octal, or decimal otherwise.
|
||
|
||
For bases up to 36, case is ignored; upper-case and lower-case
|
||
letters have the same value. For bases 37 to 62, upper-case
|
||
letter represent the usual 10..35 while lower-case letter
|
||
represent 36..61.
|
||
|
||
Return the number of bytes read, or if an error occurred, return 0.
|
||
|
||
-- Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)
|
||
Output OP on stdio stream STREAM, in raw binary format. The
|
||
integer is written in a portable format, with 4 bytes of size
|
||
information, and that many bytes of limbs. Both the size and the
|
||
limbs are written in decreasing significance order (i.e., in
|
||
big-endian).
|
||
|
||
The output can be read with `mpz_inp_raw'.
|
||
|
||
Return the number of bytes written, or if an error occurred,
|
||
return 0.
|
||
|
||
The output of this can not be read by `mpz_inp_raw' from GMP 1,
|
||
because of changes necessary for compatibility between 32-bit and
|
||
64-bit machines.
|
||
|
||
-- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
|
||
Input from stdio stream STREAM in the format written by
|
||
`mpz_out_raw', and put the result in ROP. Return the number of
|
||
bytes read, or if an error occurred, return 0.
|
||
|
||
This routine can read the output from `mpz_out_raw' also from GMP
|
||
1, in spite of changes necessary for compatibility between 32-bit
|
||
and 64-bit machines.
|
||
|
||
|
||
File: mpir.info, Node: Integer Random Numbers, Next: Integer Import and Export, Prev: I/O of Integers, Up: Integer Functions
|
||
|
||
5.13 Random Number Functions
|
||
============================
|
||
|
||
The random number functions of MPIR come in two groups; older function
|
||
that rely on a global state, and newer functions that accept a state
|
||
parameter that is read and modified. Please see the *note Random
|
||
Number Functions:: for more information on how to use and not to use
|
||
random number functions.
|
||
|
||
-- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
|
||
unsigned long int N)
|
||
Generate a uniformly distributed random integer in the range 0 to
|
||
2^N-1, inclusive.
|
||
|
||
The variable STATE must be initialized by calling one of the
|
||
`gmp_randinit' functions (*note Random State Initialization::)
|
||
before invoking this function.
|
||
|
||
-- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
|
||
mpz_t N)
|
||
Generate a uniform random integer in the range 0 to N-1, inclusive.
|
||
|
||
The variable STATE must be initialized by calling one of the
|
||
`gmp_randinit' functions (*note Random State Initialization::)
|
||
before invoking this function.
|
||
|
||
-- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
|
||
unsigned long int N)
|
||
Generate a random integer with long strings of zeros and ones in
|
||
the binary representation. Useful for testing functions and
|
||
algorithms, since this kind of random numbers have proven to be
|
||
more likely to trigger corner-case bugs. The random number will
|
||
be in the range 0 to 2^N-1, inclusive.
|
||
|
||
The variable STATE must be initialized by calling one of the
|
||
`gmp_randinit' functions (*note Random State Initialization::)
|
||
before invoking this function.
|
||
|
||
-- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
|
||
Generate a random integer of at most MAX_SIZE limbs. The generated
|
||
random number doesn't satisfy any particular requirements of
|
||
randomness. Negative random numbers are generated when MAX_SIZE
|
||
is negative.
|
||
|
||
This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm'
|
||
instead.
|
||
|
||
-- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
|
||
Generate a random integer of at most MAX_SIZE limbs, with long
|
||
strings of zeros and ones in the binary representation. Useful
|
||
for testing functions and algorithms, since this kind of random
|
||
numbers have proven to be more likely to trigger corner-case bugs.
|
||
Negative random numbers are generated when MAX_SIZE is negative.
|
||
|
||
This function is obsolete. Use `mpz_rrandomb' instead.
|
||
|
||
|
||
File: mpir.info, Node: Integer Import and Export, Next: Miscellaneous Integer Functions, Prev: Integer Random Numbers, Up: Integer Functions
|
||
|
||
5.14 Integer Import and Export
|
||
==============================
|
||
|
||
`mpz_t' variables can be converted to and from arbitrary words of binary
|
||
data with the following functions.
|
||
|
||
-- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER,
|
||
size_t SIZE, int ENDIAN, size_t NAILS, const void *OP)
|
||
Set ROP from an array of word data at OP.
|
||
|
||
The parameters specify the format of the data. COUNT many words
|
||
are read, each SIZE bytes. ORDER can be 1 for most significant
|
||
word first or -1 for least significant first. Within each word
|
||
ENDIAN can be 1 for most significant byte first, -1 for least
|
||
significant first, or 0 for the native endianness of the host CPU.
|
||
The most significant NAILS bits of each word are skipped, this can
|
||
be 0 to use the full words.
|
||
|
||
There is no sign taken from the data, ROP will simply be a positive
|
||
integer. An application can handle any sign itself, and apply it
|
||
for instance with `mpz_neg'.
|
||
|
||
There are no data alignment restrictions on OP, any address is
|
||
allowed.
|
||
|
||
Here's an example converting an array of `unsigned long' data, most
|
||
significant element first, and host byte order within each value.
|
||
|
||
unsigned long a[20];
|
||
mpz_t z;
|
||
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
|
||
|
||
This example assumes the full `sizeof' bytes are used for data in
|
||
the given type, which is usually true, and certainly true for
|
||
`unsigned long' everywhere we know of. However on Cray vector
|
||
systems it may be noted that `short' and `int' are always stored
|
||
in 8 bytes (and with `sizeof' indicating that) but use only 32 or
|
||
46 bits. The NAILS feature can account for this, by passing for
|
||
instance `8*sizeof(int)-INT_BIT'.
|
||
|
||
-- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER,
|
||
size_t SIZE, int ENDIAN, size_t NAILS, mpz_t OP)
|
||
Fill ROP with word data from OP.
|
||
|
||
The parameters specify the format of the data produced. Each word
|
||
will be SIZE bytes and ORDER can be 1 for most significant word
|
||
first or -1 for least significant first. Within each word ENDIAN
|
||
can be 1 for most significant byte first, -1 for least significant
|
||
first, or 0 for the native endianness of the host CPU. The most
|
||
significant NAILS bits of each word are unused and set to zero,
|
||
this can be 0 to produce full words.
|
||
|
||
The number of words produced is written to `*COUNTP', or COUNTP
|
||
can be `NULL' to discard the count. ROP must have enough space
|
||
for the data, or if ROP is `NULL' then a result array of the
|
||
necessary size is allocated using the current MPIR allocation
|
||
function (*note Custom Allocation::). In either case the return
|
||
value is the destination used, either ROP or the allocated block.
|
||
|
||
If OP is non-zero then the most significant word produced will be
|
||
non-zero. If OP is zero then the count returned will be zero and
|
||
nothing written to ROP. If ROP is `NULL' in this case, no block
|
||
is allocated, just `NULL' is returned.
|
||
|
||
The sign of OP is ignored, just the absolute value is exported. An
|
||
application can use `mpz_sgn' to get the sign and handle it as
|
||
desired. (*note Integer Comparisons::)
|
||
|
||
There are no data alignment restrictions on ROP, any address is
|
||
allowed.
|
||
|
||
When an application is allocating space itself the required size
|
||
can be determined with a calculation like the following. Since
|
||
`mpz_sizeinbase' always returns at least 1, `count' here will be
|
||
at least one, which avoids any portability problems with
|
||
`malloc(0)', though if `z' is zero no space at all is actually
|
||
needed (or written).
|
||
|
||
numb = 8*size - nail;
|
||
count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
|
||
p = malloc (count * size);
|
||
|
||
|
||
File: mpir.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions
|
||
|
||
5.15 Miscellaneous Functions
|
||
============================
|
||
|
||
-- Function: int mpz_fits_ulong_p (mpz_t OP)
|
||
-- Function: int mpz_fits_slong_p (mpz_t OP)
|
||
-- Function: int mpz_fits_uint_p (mpz_t OP)
|
||
-- Function: int mpz_fits_sint_p (mpz_t OP)
|
||
-- Function: int mpz_fits_ushort_p (mpz_t OP)
|
||
-- Function: int mpz_fits_sshort_p (mpz_t OP)
|
||
Return non-zero iff the value of OP fits in an `unsigned long int',
|
||
`signed long int', `unsigned int', `signed int', `unsigned short
|
||
int', or `signed short int', respectively. Otherwise, return zero.
|
||
|
||
-- Macro: int mpz_odd_p (mpz_t OP)
|
||
-- Macro: int mpz_even_p (mpz_t OP)
|
||
Determine whether OP is odd or even, respectively. Return
|
||
non-zero if yes, zero if no. These macros evaluate their argument
|
||
more than once.
|
||
|
||
-- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
|
||
Return the size of OP measured in number of digits in the given
|
||
BASE. BASE can vary from 2 to 36. The sign of OP is ignored,
|
||
just the absolute value is used. The result will be either exact
|
||
or 1 too big. If BASE is a power of 2, the result is always
|
||
exact. If OP is zero the return value is always 1.
|
||
|
||
This function can be used to determine the space required when
|
||
converting OP to a string. The right amount of allocation is
|
||
normally two more than the value returned by `mpz_sizeinbase', one
|
||
extra for a minus sign and one for the null-terminator.
|
||
|
||
It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate
|
||
the most significant 1 bit in OP, counting from 1. (Unlike the
|
||
bitwise functions which start from 0, *Note Logical and Bit
|
||
Manipulation Functions: Integer Logic and Bit Fiddling.)
|
||
|
||
|
||
File: mpir.info, Node: Integer Special Functions, Prev: Miscellaneous Integer Functions, Up: Integer Functions
|
||
|
||
5.16 Special Functions
|
||
======================
|
||
|
||
The functions in this section are for various special purposes. Most
|
||
applications will not need them.
|
||
|
||
-- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, size_t
|
||
ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
|
||
This is a special type of initialization. *Fixed* space of
|
||
FIXED_NUM_BITS is allocated to each of the ARRAY_SIZE integers in
|
||
INTEGER_ARRAY. There is no way to free the storage allocated by
|
||
this function. Don't call `mpz_clear'!
|
||
|
||
The INTEGER_ARRAY parameter is the first `mpz_t' in the array. For
|
||
example,
|
||
|
||
mpz_t arr[20000];
|
||
mpz_array_init (arr[0], 20000, 512);
|
||
|
||
This function is only intended for programs that create a large
|
||
number of integers and need to reduce memory usage by avoiding the
|
||
overheads of allocating and reallocating lots of small blocks. In
|
||
normal programs this function is not recommended.
|
||
|
||
The space allocated to each integer by this function will not be
|
||
automatically increased, unlike the normal `mpz_init', so an
|
||
application must ensure it is sufficient for any value stored.
|
||
The following space requirements apply to various routines,
|
||
|
||
* `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and
|
||
`mpz_set_ui' need room for the value they store.
|
||
|
||
* `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room
|
||
for the larger of the two operands, plus an extra
|
||
`mp_bits_per_limb'.
|
||
|
||
* `mpz_mul', `mpz_mul_ui' and `mpz_mul_ui' need room for the sum
|
||
of the number of bits in their operands, but each rounded up
|
||
to a multiple of `mp_bits_per_limb'.
|
||
|
||
* `mpz_swap' can be used between two array variables, but not
|
||
between an array and a normal variable.
|
||
|
||
For other functions, or if in doubt, the suggestion is to
|
||
calculate in a regular `mpz_init' variable and copy the result to
|
||
an array variable with `mpz_set'.
|
||
|
||
-- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
|
||
Change the space for INTEGER to NEW_ALLOC limbs. The value in
|
||
INTEGER is preserved if it fits, or is set to 0 if not. The return
|
||
value is not useful to applications and should be ignored.
|
||
|
||
`mpz_realloc2' is the preferred way to accomplish allocation
|
||
changes like this. `mpz_realloc2' and `_mpz_realloc' are the same
|
||
except that `_mpz_realloc' takes its size in limbs.
|
||
|
||
-- Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N)
|
||
Return limb number N from OP. The sign of OP is ignored, just the
|
||
absolute value is used. The least significant limb is number 0.
|
||
|
||
`mpz_size' can be used to find how many limbs make up OP.
|
||
`mpz_getlimbn' returns zero if N is outside the range 0 to
|
||
`mpz_size(OP)-1'.
|
||
|
||
-- Function: size_t mpz_size (mpz_t OP)
|
||
Return the size of OP measured in number of limbs. If OP is zero,
|
||
the returned value will be zero.
|
||
|
||
|
||
File: mpir.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
|
||
|
||
6 Rational Number Functions
|
||
***************************
|
||
|
||
This chapter describes the MPIR functions for performing arithmetic on
|
||
rational numbers. These functions start with the prefix `mpq_'.
|
||
|
||
Rational numbers are stored in objects of type `mpq_t'.
|
||
|
||
All rational arithmetic functions assume operands have a canonical
|
||
form, and canonicalize their result. The canonical from means that the
|
||
denominator and the numerator have no common factors, and that the
|
||
denominator is positive. Zero has the unique representation 0/1.
|
||
|
||
Pure assignment functions do not canonicalize the assigned variable.
|
||
It is the responsibility of the user to canonicalize the assigned
|
||
variable before any arithmetic operations are performed on that
|
||
variable.
|
||
|
||
-- Function: void mpq_canonicalize (mpq_t OP)
|
||
Remove any factors that are common to the numerator and
|
||
denominator of OP, and make the denominator positive.
|
||
|
||
* Menu:
|
||
|
||
* Initializing Rationals::
|
||
* Rational Conversions::
|
||
* Rational Arithmetic::
|
||
* Comparing Rationals::
|
||
* Applying Integer Functions::
|
||
* I/O of Rationals::
|
||
|
||
|
||
File: mpir.info, Node: Initializing Rationals, Next: Rational Conversions, Prev: Rational Number Functions, Up: Rational Number Functions
|
||
|
||
6.1 Initialization and Assignment Functions
|
||
===========================================
|
||
|
||
-- Function: void mpq_init (mpq_t DEST_RATIONAL)
|
||
Initialize DEST_RATIONAL and set it to 0/1. Each variable should
|
||
normally only be initialized once, or at least cleared out (using
|
||
the function `mpq_clear') between each initialization.
|
||
|
||
-- Function: void mpq_clear (mpq_t RATIONAL_NUMBER)
|
||
Free the space occupied by RATIONAL_NUMBER. Make sure to call this
|
||
function for all `mpq_t' variables when you are done with them.
|
||
|
||
-- Function: void mpq_set (mpq_t ROP, mpq_t OP)
|
||
-- Function: void mpq_set_z (mpq_t ROP, mpz_t OP)
|
||
Assign ROP from OP.
|
||
|
||
-- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
|
||
unsigned long int OP2)
|
||
-- Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
|
||
long int OP2)
|
||
Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have
|
||
common factors, ROP has to be passed to `mpq_canonicalize' before
|
||
any operations are performed on ROP.
|
||
|
||
-- Function: int mpq_set_str (mpq_t ROP, char *STR, int BASE)
|
||
Set ROP from a null-terminated string STR in the given BASE.
|
||
|
||
The string can be an integer like "41" or a fraction like
|
||
"41/152". The fraction must be in canonical form (*note Rational
|
||
Number Functions::), or if not then `mpq_canonicalize' must be
|
||
called.
|
||
|
||
The numerator and optional denominator are parsed the same as in
|
||
`mpz_set_str' (*note Assigning Integers::). White space is
|
||
allowed in the string, and is simply ignored. The BASE can vary
|
||
from 2 to 62, or if BASE is 0 then the leading characters are
|
||
used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for
|
||
octal, or decimal otherwise. Note that this is done separately
|
||
for the numerator and denominator, so for instance `0xEF/100' is
|
||
239/100, whereas `0xEF/0x100' is 239/256.
|
||
|
||
The return value is 0 if the entire string is a valid number, or
|
||
-1 if not.
|
||
|
||
-- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
|
||
Swap the values ROP1 and ROP2 efficiently.
|
||
|
||
|
||
File: mpir.info, Node: Rational Conversions, Next: Rational Arithmetic, Prev: Initializing Rationals, Up: Rational Number Functions
|
||
|
||
6.2 Conversion Functions
|
||
========================
|
||
|
||
-- Function: double mpq_get_d (mpq_t OP)
|
||
Convert OP to a `double', truncating if necessary (ie. rounding
|
||
towards zero).
|
||
|
||
If the exponent from the conversion is too big or too small to fit
|
||
a `double' then the result is system dependent. For too big an
|
||
infinity is returned when available. For too small 0.0 is
|
||
normally returned. Hardware overflow, underflow and denorm traps
|
||
may or may not occur.
|
||
|
||
-- Function: void mpq_set_d (mpq_t ROP, double OP)
|
||
-- Function: void mpq_set_f (mpq_t ROP, mpf_t OP)
|
||
Set ROP to the value of OP. There is no rounding, this conversion
|
||
is exact.
|
||
|
||
-- Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP)
|
||
Convert OP to a string of digits in base BASE. The base may vary
|
||
from 2 to 36. The string will be of the form `num/den', or if the
|
||
denominator is 1 then just `num'.
|
||
|
||
If STR is `NULL', the result string is allocated using the current
|
||
allocation function (*note Custom Allocation::). The block will be
|
||
`strlen(str)+1' bytes, that being exactly enough for the string and
|
||
null-terminator.
|
||
|
||
If STR is not `NULL', it should point to a block of storage large
|
||
enough for the result, that being
|
||
|
||
mpz_sizeinbase (mpq_numref(OP), BASE)
|
||
+ mpz_sizeinbase (mpq_denref(OP), BASE) + 3
|
||
|
||
The three extra bytes are for a possible minus sign, possible
|
||
slash, and the null-terminator.
|
||
|
||
A pointer to the result string is returned, being either the
|
||
allocated block, or the given STR.
|
||
|
||
|
||
File: mpir.info, Node: Rational Arithmetic, Next: Comparing Rationals, Prev: Rational Conversions, Up: Rational Number Functions
|
||
|
||
6.3 Arithmetic Functions
|
||
========================
|
||
|
||
-- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
|
||
Set SUM to ADDEND1 + ADDEND2.
|
||
|
||
-- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
|
||
SUBTRAHEND)
|
||
Set DIFFERENCE to MINUEND - SUBTRAHEND.
|
||
|
||
-- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
|
||
MULTIPLICAND)
|
||
Set PRODUCT to MULTIPLIER times MULTIPLICAND.
|
||
|
||
-- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, unsigned long
|
||
int OP2)
|
||
Set ROP to OP1 times 2 raised to OP2.
|
||
|
||
-- Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
|
||
DIVISOR)
|
||
Set QUOTIENT to DIVIDEND/DIVISOR.
|
||
|
||
-- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, unsigned long
|
||
int OP2)
|
||
Set ROP to OP1 divided by 2 raised to OP2.
|
||
|
||
-- Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
|
||
Set NEGATED_OPERAND to -OPERAND.
|
||
|
||
-- Function: void mpq_abs (mpq_t ROP, mpq_t OP)
|
||
Set ROP to the absolute value of OP.
|
||
|
||
-- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
|
||
Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
|
||
this routine will divide by zero.
|
||
|
||
|
||
File: mpir.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Rational Arithmetic, Up: Rational Number Functions
|
||
|
||
6.4 Comparison Functions
|
||
========================
|
||
|
||
-- Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)
|
||
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
|
||
if OP1 = OP2, and a negative value if OP1 < OP2.
|
||
|
||
To determine if two rationals are equal, `mpq_equal' is faster than
|
||
`mpq_cmp'.
|
||
|
||
-- Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned
|
||
long int DEN2)
|
||
-- Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int
|
||
DEN2)
|
||
Compare OP1 and NUM2/DEN2. Return a positive value if OP1 >
|
||
NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
|
||
NUM2/DEN2.
|
||
|
||
NUM2 and DEN2 are allowed to have common factors.
|
||
|
||
These functions are implemented as a macros and evaluate their
|
||
arguments multiple times.
|
||
|
||
-- Macro: int mpq_sgn (mpq_t OP)
|
||
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
|
||
|
||
This function is actually implemented as a macro. It evaluates its
|
||
arguments multiple times.
|
||
|
||
-- Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
|
||
Return non-zero if OP1 and OP2 are equal, zero if they are
|
||
non-equal. Although `mpq_cmp' can be used for the same purpose,
|
||
this function is much faster.
|
||
|
||
|
||
File: mpir.info, Node: Applying Integer Functions, Next: I/O of Rationals, Prev: Comparing Rationals, Up: Rational Number Functions
|
||
|
||
6.5 Applying Integer Functions to Rationals
|
||
===========================================
|
||
|
||
The set of `mpq' functions is quite small. In particular, there are few
|
||
functions for either input or output. The following functions give
|
||
direct access to the numerator and denominator of an `mpq_t'.
|
||
|
||
Note that if an assignment to the numerator and/or denominator could
|
||
take an `mpq_t' out of the canonical form described at the start of
|
||
this chapter (*note Rational Number Functions::) then
|
||
`mpq_canonicalize' must be called before any other `mpq' functions are
|
||
applied to that `mpq_t'.
|
||
|
||
-- Macro: mpz_t mpq_numref (mpq_t OP)
|
||
-- Macro: mpz_t mpq_denref (mpq_t OP)
|
||
Return a reference to the numerator and denominator of OP,
|
||
respectively. The `mpz' functions can be used on the result of
|
||
these macros.
|
||
|
||
-- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
|
||
-- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
|
||
-- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
|
||
-- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
|
||
Get or set the numerator or denominator of a rational. These
|
||
functions are equivalent to calling `mpz_set' with an appropriate
|
||
`mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or
|
||
`mpq_denref' is recommended instead of these functions.
|
||
|
||
|
||
File: mpir.info, Node: I/O of Rationals, Prev: Applying Integer Functions, Up: Rational Number Functions
|
||
|
||
6.6 Input and Output Functions
|
||
==============================
|
||
|
||
When using any of these functions, it's a good idea to include `stdio.h'
|
||
before `mpir.h', since that will allow `mpir.h' to define prototypes
|
||
for these functions.
|
||
|
||
Passing a `NULL' pointer for a STREAM argument to any of these
|
||
functions will make them read from `stdin' and write to `stdout',
|
||
respectively.
|
||
|
||
-- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
|
||
Output OP on stdio stream STREAM, as a string of digits in base
|
||
BASE. The base may vary from 2 to 36. Output is in the form
|
||
`num/den' or if the denominator is 1 then just `num'.
|
||
|
||
Return the number of bytes written, or if an error occurred,
|
||
return 0.
|
||
|
||
-- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
|
||
Read a string of digits from STREAM and convert them to a rational
|
||
in ROP. Any initial white-space characters are read and
|
||
discarded. Return the number of characters read (including white
|
||
space), or 0 if a rational could not be read.
|
||
|
||
The input can be a fraction like `17/63' or just an integer like
|
||
`123'. Reading stops at the first character not in this form, and
|
||
white space is not permitted within the string. If the input
|
||
might not be in canonical form, then `mpq_canonicalize' must be
|
||
called (*note Rational Number Functions::).
|
||
|
||
The BASE can be between 2 and 36, or can be 0 in which case the
|
||
leading characters of the string determine the base, `0x' or `0X'
|
||
for hexadecimal, `0' for octal, or decimal otherwise. The leading
|
||
characters are examined separately for the numerator and
|
||
denominator of a fraction, so for instance `0x10/11' is 16/11,
|
||
whereas `0x10/0x11' is 16/17.
|
||
|
||
|
||
File: mpir.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top
|
||
|
||
7 Floating-point Functions
|
||
**************************
|
||
|
||
MPIR floating point numbers are stored in objects of type `mpf_t' and
|
||
functions operating on them have an `mpf_' prefix.
|
||
|
||
The mantissa of each float has a user-selectable precision, limited
|
||
only by available memory. Each variable has its own precision, and
|
||
that can be increased or decreased at any time.
|
||
|
||
The exponent of each float is a fixed precision, one machine word on
|
||
most systems. In the current implementation the exponent is a count of
|
||
limbs, so for example on a 32-bit system this means a range of roughly
|
||
2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
|
||
greater. Note however `mpf_get_str' can only return an exponent which
|
||
fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents
|
||
bigger than a `long'.
|
||
|
||
Each variable keeps a size for the mantissa data actually in use.
|
||
This means that if a float is exactly represented in only a few bits
|
||
then only those bits will be used in a calculation, even if the
|
||
selected precision is high.
|
||
|
||
All calculations are performed to the precision of the destination
|
||
variable. Each function is defined to calculate with "infinite
|
||
precision" followed by a truncation to the destination precision, but
|
||
of course the work done is only what's needed to determine a result
|
||
under that definition.
|
||
|
||
The precision selected for a variable is a minimum value, MPIR may
|
||
increase it a little to facilitate efficient calculation. Currently
|
||
this means rounding up to a whole limb, and then sometimes having a
|
||
further partial limb, depending on the high limb of the mantissa. But
|
||
applications shouldn't be concerned by such details.
|
||
|
||
The mantissa in stored in binary, as might be imagined from the fact
|
||
precisions are expressed in bits. One consequence of this is that
|
||
decimal fractions like 0.1 cannot be represented exactly. The same is
|
||
true of plain IEEE `double' floats. This makes both highly unsuitable
|
||
for calculations involving money or other values that should be exact
|
||
decimal fractions. (Suitably scaled integers, or perhaps rationals,
|
||
are better choices.)
|
||
|
||
`mpf' functions and variables have no special notion of infinity or
|
||
not-a-number, and applications must take care not to overflow the
|
||
exponent or results will be unpredictable. This might change in a
|
||
future release.
|
||
|
||
Note that the `mpf' functions are _not_ intended as a smooth
|
||
extension to IEEE P754 arithmetic. In particular results obtained on
|
||
one computer often differ from the results on a computer with a
|
||
different word size.
|
||
|
||
* Menu:
|
||
|
||
* Initializing Floats::
|
||
* Assigning Floats::
|
||
* Simultaneous Float Init & Assign::
|
||
* Converting Floats::
|
||
* Float Arithmetic::
|
||
* Float Comparison::
|
||
* I/O of Floats::
|
||
* Miscellaneous Float Functions::
|
||
|
||
|
||
File: mpir.info, Node: Initializing Floats, Next: Assigning Floats, Prev: Floating-point Functions, Up: Floating-point Functions
|
||
|
||
7.1 Initialization Functions
|
||
============================
|
||
|
||
-- Function: void mpf_set_default_prec (unsigned long int PREC)
|
||
Set the default precision to be *at least* PREC bits. All
|
||
subsequent calls to `mpf_init' will use this precision, but
|
||
previously initialized variables are unaffected.
|
||
|
||
-- Function: unsigned long int mpf_get_default_prec (void)
|
||
Return the default precision actually used.
|
||
|
||
An `mpf_t' object must be initialized before storing the first value
|
||
in it. The functions `mpf_init' and `mpf_init2' are used for that
|
||
purpose.
|
||
|
||
-- Function: void mpf_init (mpf_t X)
|
||
Initialize X to 0. Normally, a variable should be initialized
|
||
once only or at least be cleared, using `mpf_clear', between
|
||
initializations. The precision of X is undefined unless a default
|
||
precision has already been established by a call to
|
||
`mpf_set_default_prec'.
|
||
|
||
-- Function: void mpf_init2 (mpf_t X, unsigned long int PREC)
|
||
Initialize X to 0 and set its precision to be *at least* PREC
|
||
bits. Normally, a variable should be initialized once only or at
|
||
least be cleared, using `mpf_clear', between initializations.
|
||
|
||
-- Function: void mpf_clear (mpf_t X)
|
||
Free the space occupied by X. Make sure to call this function for
|
||
all `mpf_t' variables when you are done with them.
|
||
|
||
Here is an example on how to initialize floating-point variables:
|
||
{
|
||
mpf_t x, y;
|
||
mpf_init (x); /* use default precision */
|
||
mpf_init2 (y, 256); /* precision _at least_ 256 bits */
|
||
...
|
||
/* Unless the program is about to exit, do ... */
|
||
mpf_clear (x);
|
||
mpf_clear (y);
|
||
}
|
||
|
||
The following three functions are useful for changing the precision
|
||
during a calculation. A typical use would be for adjusting the
|
||
precision gradually in iterative algorithms like Newton-Raphson, making
|
||
the computation precision closely match the actual accurate part of the
|
||
numbers.
|
||
|
||
-- Function: unsigned long int mpf_get_prec (mpf_t OP)
|
||
Return the current precision of OP, in bits.
|
||
|
||
-- Function: void mpf_set_prec (mpf_t ROP, unsigned long int PREC)
|
||
Set the precision of ROP to be *at least* PREC bits. The value in
|
||
ROP will be truncated to the new precision.
|
||
|
||
This function requires a call to `realloc', and so should not be
|
||
used in a tight loop.
|
||
|
||
-- Function: void mpf_set_prec_raw (mpf_t ROP, unsigned long int PREC)
|
||
Set the precision of ROP to be *at least* PREC bits, without
|
||
changing the memory allocated.
|
||
|
||
PREC must be no more than the allocated precision for ROP, that
|
||
being the precision when ROP was initialized, or in the most recent
|
||
`mpf_set_prec'.
|
||
|
||
The value in ROP is unchanged, and in particular if it had a higher
|
||
precision than PREC it will retain that higher precision. New
|
||
values written to ROP will use the new PREC.
|
||
|
||
Before calling `mpf_clear' or the full `mpf_set_prec', another
|
||
`mpf_set_prec_raw' call must be made to restore ROP to its original
|
||
allocated precision. Failing to do so will have unpredictable
|
||
results.
|
||
|
||
`mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
|
||
original allocated precision. After `mpf_set_prec_raw' it
|
||
reflects the PREC value set.
|
||
|
||
`mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
|
||
at different precisions during a calculation, perhaps to gradually
|
||
increase precision in an iteration, or just to use various
|
||
different precisions for different purposes during a calculation.
|
||
|
||
|
||
File: mpir.info, Node: Assigning Floats, Next: Simultaneous Float Init & Assign, Prev: Initializing Floats, Up: Floating-point Functions
|
||
|
||
7.2 Assignment Functions
|
||
========================
|
||
|
||
These functions assign new values to already initialized floats (*note
|
||
Initializing Floats::).
|
||
|
||
-- Function: void mpf_set (mpf_t ROP, mpf_t OP)
|
||
-- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
|
||
-- Function: void mpf_set_si (mpf_t ROP, signed long int OP)
|
||
-- Function: void mpf_set_d (mpf_t ROP, double OP)
|
||
-- Function: void mpf_set_z (mpf_t ROP, mpz_t OP)
|
||
-- Function: void mpf_set_q (mpf_t ROP, mpq_t OP)
|
||
Set the value of ROP from OP.
|
||
|
||
-- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
|
||
Set the value of ROP from the string in STR. The string is of the
|
||
form `M@N' or, if the base is 10 or less, alternatively `MeN'.
|
||
`M' is the mantissa and `N' is the exponent. The mantissa is
|
||
always in the specified base. The exponent is either in the
|
||
specified base or, if BASE is negative, in decimal. The decimal
|
||
point expected is taken from the current locale, on systems
|
||
providing `localeconv'.
|
||
|
||
The argument BASE may be in the ranges 2 to 62, or -62 to -2.
|
||
Negative values are used to specify that the exponent is in
|
||
decimal.
|
||
|
||
For bases up to 36, case is ignored; upper-case and lower-case
|
||
letters have the same value; for bases 37 to 62, upper-case letter
|
||
represent the usual 10..35 while lower-case letter represent
|
||
36..61.
|
||
|
||
Unlike the corresponding `mpz' function, the base will not be
|
||
determined from the leading characters of the string if BASE is 0.
|
||
This is so that numbers like `0.23' are not interpreted as octal.
|
||
|
||
White space is allowed in the string, and is simply ignored.
|
||
[This is not really true; white-space is ignored in the beginning
|
||
of the string and within the mantissa, but not in other places,
|
||
such as after a minus sign or in the exponent. We are considering
|
||
changing the definition of this function, making it fail when
|
||
there is any white-space in the input, since that makes a lot of
|
||
sense. Please tell us your opinion about this change. Do you
|
||
really want it to accept "3 14" as meaning 314 as it does now?]
|
||
|
||
This function returns 0 if the entire string is a valid number in
|
||
base BASE. Otherwise it returns -1.
|
||
|
||
-- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
|
||
Swap ROP1 and ROP2 efficiently. Both the values and the
|
||
precisions of the two variables are swapped.
|
||
|
||
|
||
File: mpir.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions
|
||
|
||
7.3 Combined Initialization and Assignment Functions
|
||
====================================================
|
||
|
||
For convenience, MPIR provides a parallel series of initialize-and-set
|
||
functions which initialize the output and then store the value there.
|
||
These functions' names have the form `mpf_init_set...'
|
||
|
||
Once the float has been initialized by any of the `mpf_init_set...'
|
||
functions, it can be used as the source or destination operand for the
|
||
ordinary float functions. Don't use an initialize-and-set function on
|
||
a variable already initialized!
|
||
|
||
-- Function: void mpf_init_set (mpf_t ROP, mpf_t OP)
|
||
-- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
|
||
-- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
|
||
-- Function: void mpf_init_set_d (mpf_t ROP, double OP)
|
||
Initialize ROP and set its value from OP.
|
||
|
||
The precision of ROP will be taken from the active default
|
||
precision, as set by `mpf_set_default_prec'.
|
||
|
||
-- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
|
||
Initialize ROP and set its value from the string in STR. See
|
||
`mpf_set_str' above for details on the assignment operation.
|
||
|
||
Note that ROP is initialized even if an error occurs. (I.e., you
|
||
have to call `mpf_clear' for it.)
|
||
|
||
The precision of ROP will be taken from the active default
|
||
precision, as set by `mpf_set_default_prec'.
|
||
|
||
|
||
File: mpir.info, Node: Converting Floats, Next: Float Arithmetic, Prev: Simultaneous Float Init & Assign, Up: Floating-point Functions
|
||
|
||
7.4 Conversion Functions
|
||
========================
|
||
|
||
-- Function: double mpf_get_d (mpf_t OP)
|
||
Convert OP to a `double', truncating if necessary (ie. rounding
|
||
towards zero).
|
||
|
||
If the exponent in OP is too big or too small to fit a `double'
|
||
then the result is system dependent. For too big an infinity is
|
||
returned when available. For too small 0.0 is normally returned.
|
||
Hardware overflow, underflow and denorm traps may or may not occur.
|
||
|
||
-- Function: double mpf_get_d_2exp (signed long int *EXP, mpf_t OP)
|
||
Convert OP to a `double', truncating if necessary (ie. rounding
|
||
towards zero), and with an exponent returned separately.
|
||
|
||
The return value is in the range 0.5<=abs(D)<1 and the exponent is
|
||
stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
|
||
is zero, the return is 0.0 and 0 is stored to `*EXP'.
|
||
|
||
This is similar to the standard C `frexp' function (*note
|
||
Normalization Functions: (libc)Normalization Functions.).
|
||
|
||
-- Function: long mpf_get_si (mpf_t OP)
|
||
-- Function: unsigned long mpf_get_ui (mpf_t OP)
|
||
Convert OP to a `long' or `unsigned long', truncating any fraction
|
||
part. If OP is too big for the return type, the result is
|
||
undefined.
|
||
|
||
See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
|
||
Miscellaneous Float Functions::).
|
||
|
||
-- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int
|
||
BASE, size_t N_DIGITS, mpf_t OP)
|
||
Convert OP to a string of digits in base BASE. BASE can be 2 to
|
||
36. Up to N_DIGITS digits will be generated. Trailing zeros are
|
||
not returned. No more digits than can be accurately represented
|
||
by OP are ever generated. If N_DIGITS is 0 then that accurate
|
||
maximum number of digits are generated.
|
||
|
||
If STR is `NULL', the result string is allocated using the current
|
||
allocation function (*note Custom Allocation::). The block will be
|
||
`strlen(str)+1' bytes, that being exactly enough for the string and
|
||
null-terminator.
|
||
|
||
If STR is not `NULL', it should point to a block of N_DIGITS + 2
|
||
bytes, that being enough for the mantissa, a possible minus sign,
|
||
and a null-terminator. When N_DIGITS is 0 to get all significant
|
||
digits, an application won't be able to know the space required,
|
||
and STR should be `NULL' in that case.
|
||
|
||
The generated string is a fraction, with an implicit radix point
|
||
immediately to the left of the first digit. The applicable
|
||
exponent is written through the EXPPTR pointer. For example, the
|
||
number 3.1416 would be returned as string "31416" and exponent 1.
|
||
|
||
When OP is zero, an empty string is produced and the exponent
|
||
returned is 0.
|
||
|
||
A pointer to the result string is returned, being either the
|
||
allocated block or the given STR.
|
||
|
||
|
||
File: mpir.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions
|
||
|
||
7.5 Arithmetic Functions
|
||
========================
|
||
|
||
-- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
|
||
-- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int
|
||
OP2)
|
||
Set ROP to OP1 + OP2.
|
||
|
||
-- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
|
||
-- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t
|
||
OP2)
|
||
-- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int
|
||
OP2)
|
||
Set ROP to OP1 - OP2.
|
||
|
||
-- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
|
||
-- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int
|
||
OP2)
|
||
Set ROP to OP1 times OP2.
|
||
|
||
Division is undefined if the divisor is zero, and passing a zero
|
||
divisor to the divide functions will make these functions intentionally
|
||
divide by zero. This lets the user handle arithmetic exceptions in
|
||
these functions in the same manner as other arithmetic exceptions.
|
||
|
||
-- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
|
||
-- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t
|
||
OP2)
|
||
-- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int
|
||
OP2)
|
||
Set ROP to OP1/OP2.
|
||
|
||
-- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
|
||
-- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
|
||
Set ROP to the square root of OP.
|
||
|
||
-- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int
|
||
OP2)
|
||
Set ROP to OP1 raised to the power OP2.
|
||
|
||
-- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
|
||
Set ROP to -OP.
|
||
|
||
-- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
|
||
Set ROP to the absolute value of OP.
|
||
|
||
-- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, unsigned long
|
||
int OP2)
|
||
Set ROP to OP1 times 2 raised to OP2.
|
||
|
||
-- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, unsigned long
|
||
int OP2)
|
||
Set ROP to OP1 divided by 2 raised to OP2.
|
||
|
||
|
||
File: mpir.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions
|
||
|
||
7.6 Comparison Functions
|
||
========================
|
||
|
||
-- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
|
||
-- Function: int mpf_cmp_d (mpf_t OP1, double OP2)
|
||
-- Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)
|
||
-- Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)
|
||
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
|
||
if OP1 = OP2, and a negative value if OP1 < OP2.
|
||
|
||
`mpf_cmp_d' can be called with an infinity, but results are
|
||
undefined for a NaN.
|
||
|
||
-- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, unsigned long int op3)
|
||
Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
|
||
zero otherwise. I.e., test if OP1 and OP2 are approximately equal.
|
||
|
||
Caution: Currently only whole limbs are compared, and only in an
|
||
exact fashion. In the future values like 1000 and 0111 may be
|
||
considered the same to 3 bits (on the basis that their difference
|
||
is that small).
|
||
|
||
-- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
|
||
Compute the relative difference between OP1 and OP2 and store the
|
||
result in ROP. This is abs(OP1-OP2)/OP1.
|
||
|
||
-- Macro: int mpf_sgn (mpf_t OP)
|
||
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
|
||
|
||
This function is actually implemented as a macro. It evaluates
|
||
its arguments multiple times.
|
||
|
||
|
||
File: mpir.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions
|
||
|
||
7.7 Input and Output Functions
|
||
==============================
|
||
|
||
Functions that perform input from a stdio stream, and functions that
|
||
output to a stdio stream. Passing a `NULL' pointer for a STREAM
|
||
argument to any of these functions will make them read from `stdin' and
|
||
write to `stdout', respectively.
|
||
|
||
When using any of these functions, it is a good idea to include
|
||
`stdio.h' before `mpir.h', since that will allow `mpir.h' to define
|
||
prototypes for these functions.
|
||
|
||
-- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
|
||
N_DIGITS, mpf_t OP)
|
||
Print OP to STREAM, as a string of digits. Return the number of
|
||
bytes written, or if an error occurred, return 0.
|
||
|
||
The mantissa is prefixed with an `0.' and is in the given BASE,
|
||
which may vary from 2 to 36. An exponent then printed, separated
|
||
by an `e', or if BASE is greater than 10 then by an `@'. The
|
||
exponent is always in decimal. The decimal point follows the
|
||
current locale, on systems providing `localeconv'.
|
||
|
||
Up to N_DIGITS will be printed from the mantissa, except that no
|
||
more digits than are accurately representable by OP will be
|
||
printed. N_DIGITS can be 0 to select that accurate maximum.
|
||
|
||
-- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
|
||
Read a string in base BASE from STREAM, and put the read float in
|
||
ROP. The string is of the form `M@N' or, if the base is 10 or
|
||
less, alternatively `MeN'. `M' is the mantissa and `N' is the
|
||
exponent. The mantissa is always in the specified base. The
|
||
exponent is either in the specified base or, if BASE is negative,
|
||
in decimal. The decimal point expected is taken from the current
|
||
locale, on systems providing `localeconv'.
|
||
|
||
The argument BASE may be in the ranges 2 to 36, or -36 to -2.
|
||
Negative values are used to specify that the exponent is in
|
||
decimal.
|
||
|
||
Unlike the corresponding `mpz' function, the base will not be
|
||
determined from the leading characters of the string if BASE is 0.
|
||
This is so that numbers like `0.23' are not interpreted as octal.
|
||
|
||
Return the number of bytes read, or if an error occurred, return 0.
|
||
|
||
|
||
File: mpir.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions
|
||
|
||
7.8 Miscellaneous Functions
|
||
===========================
|
||
|
||
-- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
|
||
-- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
|
||
-- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
|
||
Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
|
||
next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
|
||
to the integer towards zero.
|
||
|
||
-- Function: int mpf_integer_p (mpf_t OP)
|
||
Return non-zero if OP is an integer.
|
||
|
||
-- Function: int mpf_fits_ulong_p (mpf_t OP)
|
||
-- Function: int mpf_fits_slong_p (mpf_t OP)
|
||
-- Function: int mpf_fits_uint_p (mpf_t OP)
|
||
-- Function: int mpf_fits_sint_p (mpf_t OP)
|
||
-- Function: int mpf_fits_ushort_p (mpf_t OP)
|
||
-- Function: int mpf_fits_sshort_p (mpf_t OP)
|
||
Return non-zero if OP would fit in the respective C data type, when
|
||
truncated to an integer.
|
||
|
||
-- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
|
||
unsigned long int NBITS)
|
||
Generate a uniformly distributed random float in ROP, such that 0
|
||
<= ROP < 1, with NBITS significant bits in the mantissa.
|
||
|
||
The variable STATE must be initialized by calling one of the
|
||
`gmp_randinit' functions (*note Random State Initialization::)
|
||
before invoking this function.
|
||
|
||
-- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
|
||
EXP)
|
||
Generate a random float of at most MAX_SIZE limbs, with long
|
||
strings of zeros and ones in the binary representation. The
|
||
exponent of the number is in the interval -EXP to EXP (in limbs).
|
||
This function is useful for testing functions and algorithms,
|
||
since these kind of random numbers have proven to be more likely
|
||
to trigger corner-case bugs. Negative random numbers are
|
||
generated when MAX_SIZE is negative.
|
||
|
||
|
||
File: mpir.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
|
||
|
||
8 Low-level Functions
|
||
*********************
|
||
|
||
This chapter describes low-level MPIR functions, used to implement the
|
||
high-level MPIR functions, but also intended for time-critical user
|
||
code.
|
||
|
||
These functions start with the prefix `mpn_'.
|
||
|
||
The `mpn' functions are designed to be as fast as possible, *not* to
|
||
provide a coherent calling interface. The different functions have
|
||
somewhat similar interfaces, but there are variations that make them
|
||
hard to use. These functions do as little as possible apart from the
|
||
real multiple precision computation, so that no time is spent on things
|
||
that not all callers need.
|
||
|
||
A source operand is specified by a pointer to the least significant
|
||
limb and a limb count. A destination operand is specified by just a
|
||
pointer. It is the responsibility of the caller to ensure that the
|
||
destination has enough space for storing the result.
|
||
|
||
With this way of specifying operands, it is possible to perform
|
||
computations on subranges of an argument, and store the result into a
|
||
subrange of a destination.
|
||
|
||
A common requirement for all functions is that each source area
|
||
needs at least one limb. No size argument may be zero. Unless
|
||
otherwise stated, in-place operations are allowed where source and
|
||
destination are the same, but not where they only partly overlap.
|
||
|
||
The `mpn' functions are the base for the implementation of the
|
||
`mpz_', `mpf_', and `mpq_' functions.
|
||
|
||
This example adds the number beginning at S1P and the number
|
||
beginning at S2P and writes the sum at DESTP. All areas have N limbs.
|
||
|
||
cy = mpn_add_n (destp, s1p, s2p, n)
|
||
|
||
It should be noted that the `mpn' functions make no attempt to
|
||
identify high or low zero limbs on their operands, or other special
|
||
forms. On random data such cases will be unlikely and it'd be wasteful
|
||
for every function to check every time. An application knowing
|
||
something about its data can take steps to trim or perhaps split its
|
||
calculations.
|
||
|
||
|
||
In the notation used below, a source operand is identified by the
|
||
pointer to the least significant limb, and the limb count in braces.
|
||
For example, {S1P, S1N}.
|
||
|
||
-- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
const mp_limb_t *S2P, mp_size_t N)
|
||
Add {S1P, N} and {S2P, N}, and write the N least significant limbs
|
||
of the result to RP. Return carry, either 0 or 1.
|
||
|
||
This is the lowest-level function for addition. It is the
|
||
preferred function for addition, since it is written in assembly
|
||
for most CPUs. For addition of a variable to itself (i.e., S1P
|
||
equals S2P, use `mpn_lshift' with a count of 1 for optimal speed.
|
||
|
||
-- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
mp_size_t N, mp_limb_t S2LIMB)
|
||
Add {S1P, N} and S2LIMB, and write the N least significant limbs
|
||
of the result to RP. Return carry, either 0 or 1.
|
||
|
||
-- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
|
||
Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
|
||
limbs of the result to RP. Return carry, either 0 or 1.
|
||
|
||
This function requires that S1N is greater than or equal to S2N.
|
||
|
||
-- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
const mp_limb_t *S2P, mp_size_t N)
|
||
Subtract {S2P, N} from {S1P, N}, and write the N least significant
|
||
limbs of the result to RP. Return borrow, either 0 or 1.
|
||
|
||
This is the lowest-level function for subtraction. It is the
|
||
preferred function for subtraction, since it is written in
|
||
assembly for most CPUs.
|
||
|
||
-- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
mp_size_t N, mp_limb_t S2LIMB)
|
||
Subtract S2LIMB from {S1P, N}, and write the N least significant
|
||
limbs of the result to RP. Return borrow, either 0 or 1.
|
||
|
||
-- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
|
||
Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
|
||
significant limbs of the result to RP. Return borrow, either 0 or
|
||
1.
|
||
|
||
This function requires that S1N is greater than or equal to S2N.
|
||
|
||
-- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
const mp_limb_t *S2P, mp_size_t N)
|
||
Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
|
||
RP.
|
||
|
||
The destination has to have space for 2*N limbs, even if the
|
||
product's most significant limb is zero. No overlap is permitted
|
||
between the destination and either source.
|
||
|
||
-- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
mp_size_t N, mp_limb_t S2LIMB)
|
||
Multiply {S1P, N} by S2LIMB, and write the N least significant
|
||
limbs of the product to RP. Return the most significant limb of
|
||
the product. {S1P, N} and {RP, N} are allowed to overlap provided
|
||
RP <= S1P.
|
||
|
||
This is a low-level function that is a building block for general
|
||
multiplication as well as other operations in MPIR. It is written
|
||
in assembly for most CPUs.
|
||
|
||
Don't call this function if S2LIMB is a power of 2; use
|
||
`mpn_lshift' with a count equal to the logarithm of S2LIMB
|
||
instead, for optimal speed.
|
||
|
||
-- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
|
||
*S1P, mp_size_t N, mp_limb_t S2LIMB)
|
||
Multiply {S1P, N} and S2LIMB, and add the N least significant
|
||
limbs of the product to {RP, N} and write the result to RP.
|
||
Return the most significant limb of the product, plus carry-out
|
||
from the addition.
|
||
|
||
This is a low-level function that is a building block for general
|
||
multiplication as well as other operations in MPIR. It is written
|
||
in assembly for most CPUs.
|
||
|
||
-- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
|
||
*S1P, mp_size_t N, mp_limb_t S2LIMB)
|
||
Multiply {S1P, N} and S2LIMB, and subtract the N least significant
|
||
limbs of the product from {RP, N} and write the result to RP.
|
||
Return the most significant limb of the product, minus borrow-out
|
||
from the subtraction.
|
||
|
||
This is a low-level function that is a building block for general
|
||
multiplication and division as well as other operations in MPIR.
|
||
It is written in assembly for most CPUs.
|
||
|
||
-- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
|
||
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
|
||
Multiply {S1P, S1N} and {S2P, S2N}, and write the result to RP.
|
||
Return the most significant limb of the result.
|
||
|
||
The destination has to have space for S1N + S2N limbs, even if the
|
||
result might be one limb smaller.
|
||
|
||
This function requires that S1N is greater than or equal to S2N.
|
||
The destination must be distinct from both input operands.
|
||
|
||
-- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
|
||
QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
|
||
mp_size_t DN)
|
||
Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
|
||
and the remainder at {RP, DN}. The quotient is rounded towards 0.
|
||
|
||
No overlap is permitted between arguments. NN must be greater
|
||
than or equal to DN. The most significant limb of DP must be
|
||
non-zero. The QXN operand must be zero.
|
||
|
||
-- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
|
||
mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
|
||
mp_size_t S3N)
|
||
[This function is obsolete. Please call `mpn_tdiv_qr' instead for
|
||
best performance.]
|
||
|
||
Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
|
||
with the exception of the most significant limb, which is
|
||
returned. The remainder replaces the dividend at RS2P; it will be
|
||
S3N limbs long (i.e., as many limbs as the divisor).
|
||
|
||
In addition to an integer quotient, QXN fraction limbs are
|
||
developed, and stored after the integral limbs. For most usages,
|
||
QXN will be zero.
|
||
|
||
It is required that RS2N is greater than or equal to S3N. It is
|
||
required that the most significant bit of the divisor is set.
|
||
|
||
If the quotient is not needed, pass RS2P + S3N as R1P. Aside from
|
||
that special case, no overlap between arguments is permitted.
|
||
|
||
Return the most significant limb of the quotient, either 0 or 1.
|
||
|
||
The area at R1P needs to be RS2N - S3N + QXN limbs large.
|
||
|
||
-- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
|
||
mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
|
||
-- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
|
||
mp_size_t S2N, mp_limb_t S3LIMB)
|
||
Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
|
||
Return the remainder.
|
||
|
||
The integer quotient is written to {R1P+QXN, S2N} and in addition
|
||
QXN fraction limbs are developed and written to {R1P, QXN}.
|
||
Either or both S2N and QXN can be zero. For most usages, QXN will
|
||
be zero.
|
||
|
||
`mpn_divmod_1' exists for upward source compatibility and is
|
||
simply a macro calling `mpn_divrem_1' with a QXN of 0.
|
||
|
||
The areas at R1P and S2P have to be identical or completely
|
||
separate, not partially overlapping.
|
||
|
||
-- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
|
||
mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N)
|
||
[This function is obsolete. Please call `mpn_tdiv_qr' instead for
|
||
best performance.]
|
||
|
||
-- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
|
||
mp_size_t N)
|
||
-- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t
|
||
*SP, mp_size_t N, mp_limb_t CARRY)
|
||
Divide {SP, N} by 3, expecting it to divide exactly, and writing
|
||
the result to {RP, N}. If 3 divides exactly, the return value is
|
||
zero and the result is the quotient. If not, the return value is
|
||
non-zero and the result won't be anything useful.
|
||
|
||
`mpn_divexact_by3c' takes an initial carry parameter, which can be
|
||
the return value from a previous call, so a large calculation can
|
||
be done piece by piece from low to high. `mpn_divexact_by3' is
|
||
simply a macro calling `mpn_divexact_by3c' with a 0 carry
|
||
parameter.
|
||
|
||
These routines use a multiply-by-inverse and will be faster than
|
||
`mpn_divrem_1' on CPUs with fast multiplication but slow division.
|
||
|
||
The source a, result q, size n, initial carry i, and return value
|
||
c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return
|
||
c is always 0, 1 or 2, and the initial carry i must also be 0, 1
|
||
or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3.
|
||
When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b ==
|
||
1 mod 3 (when `mp_bits_per_limb' is even, which is always so
|
||
currently).
|
||
|
||
-- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
|
||
mp_limb_t S2LIMB)
|
||
Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be
|
||
zero.
|
||
|
||
-- Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P,
|
||
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N, unsigned
|
||
long int D)
|
||
This function puts the low floor(D/mp_bits_per_limb) limbs of Q =
|
||
{S1P, S1N}/{S2P, S2N} mod 2^D at RP, and returns the high D mod
|
||
`mp_bits_per_limb' bits of Q.
|
||
|
||
{S1P, S1N} - Q * {S2P, S2N} mod 2^(S1N*mp_bits_per_limb) is placed
|
||
at S1P. Since the low floor(D/mp_bits_per_limb) limbs of this
|
||
difference are zero, it is possible to overwrite the low limbs at
|
||
S1P with this difference, provided RP <= S1P.
|
||
|
||
This function requires that S1N * mp_bits_per_limb >= D, and that
|
||
{S2P, S2N} is odd.
|
||
|
||
*This interface is preliminary. It might change incompatibly in
|
||
future revisions.*
|
||
|
||
-- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
|
||
mp_size_t N, unsigned int COUNT)
|
||
Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
|
||
The bits shifted out at the left are returned in the least
|
||
significant COUNT bits of the return value (the rest of the return
|
||
value is zero).
|
||
|
||
COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
|
||
{SP, N} and {RP, N} may overlap, provided RP >= SP.
|
||
|
||
This function is written in assembly for most CPUs.
|
||
|
||
-- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
|
||
mp_size_t N, unsigned int COUNT)
|
||
Shift {SP, N} right by COUNT bits, and write the result to {RP,
|
||
N}. The bits shifted out at the right are returned in the most
|
||
significant COUNT bits of the return value (the rest of the return
|
||
value is zero).
|
||
|
||
COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
|
||
{SP, N} and {RP, N} may overlap, provided RP <= SP.
|
||
|
||
This function is written in assembly for most CPUs.
|
||
|
||
-- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
|
||
mp_size_t N)
|
||
Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
|
||
S2, 0 if they are equal, or a negative value if S1 < S2.
|
||
|
||
-- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P,
|
||
mp_size_t S1N, mp_limb_t *S2P, mp_size_t S2N)
|
||
Set {RP, RETVAL} to the greatest common divisor of {S1P, S1N} and
|
||
{S2P, S2N}. The result can be up to S2N limbs, the return value
|
||
is the actual number produced. Both source operands are destroyed.
|
||
|
||
{S1P, S1N} must have at least as many bits as {S2P, S2N}. {S2P,
|
||
S2N} must be odd. Both operands must have non-zero most
|
||
significant limbs. No overlap is permitted between {S1P, S1N} and
|
||
{S2P, S2N}.
|
||
|
||
-- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t S1N,
|
||
mp_limb_t S2LIMB)
|
||
Return the greatest common divisor of {S1P, S1N} and S2LIMB. Both
|
||
operands must be non-zero.
|
||
|
||
-- Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P,
|
||
mp_size_t *R2N, mp_limb_t *S1P, mp_size_t S1N, mp_limb_t
|
||
*S2P, mp_size_t S2N)
|
||
Calculate the greatest common divisor of {S1P, S1N} and {S2P,
|
||
S2N}. Store the gcd at {R1P, RETVAL} and the first cofactor at
|
||
{R2P, *R2N}, with *R2N negative if the cofactor is negative. R1P
|
||
and R2P should each have room for S1N+1 limbs, but the return
|
||
value and value stored through R2N indicate the actual number
|
||
produced.
|
||
|
||
{S1P, S1N} >= {S2P, S2N} is required, and both must be non-zero.
|
||
The regions {S1P, S1N+1} and {S2P, S2N+1} are destroyed (i.e. the
|
||
operands plus an extra limb past the end of each).
|
||
|
||
The cofactor R1 will satisfy R2*S1 + K*S2 = R1. The second
|
||
cofactor K is not calculated but can easily be obtained from (R1 -
|
||
R2*S1) / S2.
|
||
|
||
-- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
|
||
const mp_limb_t *SP, mp_size_t N)
|
||
Compute the square root of {SP, N} and put the result at {R1P,
|
||
ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space
|
||
for N limbs, but the return value indicates how many are produced.
|
||
|
||
The most significant limb of {SP, N} must be non-zero. The areas
|
||
{R1P, ceil(N/2)} and {SP, N} must be completely separate. The
|
||
areas {R2P, N} and {SP, N} must be either identical or completely
|
||
separate.
|
||
|
||
If the remainder is not wanted then R2P can be `NULL', and in this
|
||
case the return value is zero or non-zero according to whether the
|
||
remainder would have been zero or non-zero.
|
||
|
||
A return value of zero indicates a perfect square. See also
|
||
`mpz_perfect_square_p'.
|
||
|
||
-- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
|
||
mp_limb_t *S1P, mp_size_t S1N)
|
||
Convert {S1P, S1N} to a raw unsigned char array at STR in base
|
||
BASE, and return the number of characters produced. There may be
|
||
leading zeros in the string. The string is not in ASCII; to
|
||
convert it to printable format, add the ASCII codes for `0' or
|
||
`A', depending on the base and range. BASE can vary from 2 to 256.
|
||
|
||
The most significant limb of the input {S1P, S1N} must be
|
||
non-zero. The input {S1P, S1N} is clobbered, except when BASE is
|
||
a power of 2, in which case it's unchanged.
|
||
|
||
The area at STR has to have space for the largest possible number
|
||
represented by a S1N long limb array, plus one extra character.
|
||
|
||
-- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
|
||
*STR, size_t STRSIZE, int BASE)
|
||
Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
|
||
|
||
STR[0] is the most significant byte and STR[STRSIZE-1] is the
|
||
least significant. Each byte should be a value in the range 0 to
|
||
BASE-1, not an ASCII character. BASE can vary from 2 to 256.
|
||
|
||
The return value is the number of limbs written to RP. If the most
|
||
significant input byte is non-zero then the high limb at RP will be
|
||
non-zero, and only that exact number of limbs will be required
|
||
there.
|
||
|
||
If the most significant input byte is zero then there may be high
|
||
zero limbs written to RP and included in the return value.
|
||
|
||
STRSIZE must be at least 1, and no overlap is permitted between
|
||
{STR,STRSIZE} and the result at RP.
|
||
|
||
-- Function: unsigned long int mpn_scan0 (const mp_limb_t *S1P,
|
||
unsigned long int BIT)
|
||
Scan S1P from bit position BIT for the next clear bit.
|
||
|
||
It is required that there be a clear bit within the area at S1P at
|
||
or beyond bit position BIT, so that the function has something to
|
||
return.
|
||
|
||
-- Function: unsigned long int mpn_scan1 (const mp_limb_t *S1P,
|
||
unsigned long int BIT)
|
||
Scan S1P from bit position BIT for the next set bit.
|
||
|
||
It is required that there be a set bit within the area at S1P at or
|
||
beyond bit position BIT, so that the function has something to
|
||
return.
|
||
|
||
-- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
|
||
-- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
|
||
Generate a random number of length R1N and store it at R1P. The
|
||
most significant limb is always non-zero. `mpn_random' generates
|
||
uniformly distributed limb data, `mpn_random2' generates long
|
||
strings of zeros and ones in the binary representation.
|
||
|
||
`mpn_random2' is intended for testing the correctness of the `mpn'
|
||
routines.
|
||
|
||
-- Function: unsigned long int mpn_popcount (const mp_limb_t *S1P,
|
||
mp_size_t N)
|
||
Count the number of set bits in {S1P, N}.
|
||
|
||
-- Function: unsigned long int mpn_hamdist (const mp_limb_t *S1P,
|
||
const mp_limb_t *S2P, mp_size_t N)
|
||
Compute the hamming distance between {S1P, N} and {S2P, N}, which
|
||
is the number of bit positions where the two operands have
|
||
different bit values.
|
||
|
||
-- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
|
||
N)
|
||
Return non-zero iff {S1P, N} is a perfect square.
|
||
|
||
|
||
8.1 Nails
|
||
=========
|
||
|
||
*Everything in this section is highly experimental and may disappear or
|
||
be subject to incompatible changes in a future version of MPIR.*
|
||
|
||
N.B: Nails are currently disabled and not supported in MPIR. They
|
||
may or may not return in a future version of MPIR.
|
||
|
||
Nails are an experimental feature whereby a few bits are left unused
|
||
at the top of each `mp_limb_t'. This can significantly improve carry
|
||
handling on some processors.
|
||
|
||
All the `mpn' functions accepting limb data will expect the nail
|
||
bits to be zero on entry, and will return data with the nails similarly
|
||
all zero. This applies both to limb vectors and to single limb
|
||
arguments.
|
||
|
||
Nails can be enabled by configuring with `--enable-nails'. By
|
||
default the number of bits will be chosen according to what suits the
|
||
host processor, but a particular number can be selected with
|
||
`--enable-nails=N'.
|
||
|
||
At the mpn level, a nail build is neither source nor binary
|
||
compatible with a non-nail build, strictly speaking. But programs
|
||
acting on limbs only through the mpn functions are likely to work
|
||
equally well with either build, and judicious use of the definitions
|
||
below should make any program compatible with either build, at the
|
||
source level.
|
||
|
||
For the higher level routines, meaning `mpz' etc, a nail build
|
||
should be fully source and binary compatible with a non-nail build.
|
||
|
||
-- Macro: GMP_NAIL_BITS
|
||
-- Macro: GMP_NUMB_BITS
|
||
-- Macro: GMP_LIMB_BITS
|
||
`GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are
|
||
not in use. `GMP_NUMB_BITS' is the number of data bits in a limb.
|
||
`GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'. In
|
||
all cases
|
||
|
||
GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
|
||
|
||
-- Macro: GMP_NAIL_MASK
|
||
-- Macro: GMP_NUMB_MASK
|
||
Bit masks for the nail and number parts of a limb.
|
||
`GMP_NAIL_MASK' is 0 when nails are not in use.
|
||
|
||
`GMP_NAIL_MASK' is not often needed, since the nail part can be
|
||
obtained with `x >> GMP_NUMB_BITS', and that means one less large
|
||
constant, which can help various RISC chips.
|
||
|
||
-- Macro: GMP_NUMB_MAX
|
||
The maximum value that can be stored in the number part of a limb.
|
||
This is the same as `GMP_NUMB_MASK', but can be used for clarity
|
||
when doing comparisons rather than bit-wise operations.
|
||
|
||
The term "nails" comes from finger or toe nails, which are at the
|
||
ends of a limb (arm or leg). "numb" is short for number, but is also
|
||
how the developers felt after trying for a long time to come up with
|
||
sensible names for these things.
|
||
|
||
In the future (the distant future most likely) a non-zero nail might
|
||
be permitted, giving non-unique representations for numbers in a limb
|
||
vector. This would help vector processors since carries would only
|
||
ever need to propagate one or two limbs.
|
||
|
||
|
||
File: mpir.info, Node: Random Number Functions, Next: Formatted Output, Prev: Low-level Functions, Up: Top
|
||
|
||
9 Random Number Functions
|
||
*************************
|
||
|
||
Sequences of pseudo-random numbers in MPIR are generated using a
|
||
variable of type `gmp_randstate_t', which holds an algorithm selection
|
||
and a current state. Such a variable must be initialized by a call to
|
||
one of the `gmp_randinit' functions, and can be seeded with one of the
|
||
`gmp_randseed' functions.
|
||
|
||
The functions actually generating random numbers are described in
|
||
*note Integer Random Numbers::, and *note Miscellaneous Float
|
||
Functions::.
|
||
|
||
The older style random number functions don't accept a
|
||
`gmp_randstate_t' parameter but instead share a global variable of that
|
||
type. They use a default algorithm and are currently not seeded
|
||
(though perhaps that will change in the future). The new functions
|
||
accepting a `gmp_randstate_t' are recommended for applications that
|
||
care about randomness.
|
||
|
||
* Menu:
|
||
|
||
* Random State Initialization::
|
||
* Random State Seeding::
|
||
* Random State Miscellaneous::
|
||
|
||
|
||
File: mpir.info, Node: Random State Initialization, Next: Random State Seeding, Prev: Random Number Functions, Up: Random Number Functions
|
||
|
||
9.1 Random State Initialization
|
||
===============================
|
||
|
||
-- Function: void gmp_randinit_default (gmp_randstate_t STATE)
|
||
Initialize STATE with a default algorithm. This will be a
|
||
compromise between speed and randomness, and is recommended for
|
||
applications with no special requirements. Currently this is
|
||
`gmp_randinit_mt'.
|
||
|
||
-- Function: void gmp_randinit_mt (gmp_randstate_t STATE)
|
||
Initialize STATE for a Mersenne Twister algorithm. This algorithm
|
||
is fast and has good randomness properties.
|
||
|
||
-- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t
|
||
A, unsigned long C, unsigned long M2EXP)
|
||
Initialize STATE with a linear congruential algorithm X = (A*X +
|
||
C) mod 2^M2EXP.
|
||
|
||
The low bits of X in this algorithm are not very random. The least
|
||
significant bit will have a period no more than 2, and the second
|
||
bit no more than 4, etc. For this reason only the high half of
|
||
each X is actually used.
|
||
|
||
When a random number of more than M2EXP/2 bits is to be generated,
|
||
multiple iterations of the recurrence are used and the results
|
||
concatenated.
|
||
|
||
-- Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE,
|
||
unsigned long SIZE)
|
||
Initialize STATE for a linear congruential algorithm as per
|
||
`gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table,
|
||
chosen so that SIZE bits (or more) of each X will be used, ie.
|
||
M2EXP/2 >= SIZE.
|
||
|
||
If successful the return value is non-zero. If SIZE is bigger
|
||
than the table data provides then the return value is zero. The
|
||
maximum SIZE currently supported is 128.
|
||
|
||
-- Function: int gmp_randinit_set (gmp_randstate_t ROP,
|
||
gmp_randstate_t OP)
|
||
Initialize ROP with a copy of the algorithm and state from OP.
|
||
|
||
-- Function: void gmp_randinit (gmp_randstate_t STATE,
|
||
gmp_randalg_t ALG, ...)
|
||
*This function is obsolete.*
|
||
|
||
Initialize STATE with an algorithm selected by ALG. The only
|
||
choice is `GMP_RAND_ALG_LC', which is `gmp_randinit_lc_2exp_size'
|
||
described above. A third parameter of type `unsigned long' is
|
||
required, this is the SIZE for that function.
|
||
`GMP_RAND_ALG_DEFAULT' or 0 are the same as `GMP_RAND_ALG_LC'.
|
||
|
||
`gmp_randinit' sets bits in the global variable `gmp_errno' to
|
||
indicate an error. `GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is
|
||
unsupported, or `GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter
|
||
is too big. It may be noted this error reporting is not thread
|
||
safe (a good reason to use `gmp_randinit_lc_2exp_size' instead).
|
||
|
||
-- Function: void gmp_randclear (gmp_randstate_t STATE)
|
||
Free all memory occupied by STATE.
|
||
|
||
|
||
File: mpir.info, Node: Random State Seeding, Next: Random State Miscellaneous, Prev: Random State Initialization, Up: Random Number Functions
|
||
|
||
9.2 Random State Seeding
|
||
========================
|
||
|
||
-- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
|
||
-- Function: void gmp_randseed_ui (gmp_randstate_t STATE,
|
||
unsigned long int SEED)
|
||
Set an initial seed value into STATE.
|
||
|
||
The size of a seed determines how many different sequences of
|
||
random numbers that it's possible to generate. The "quality" of
|
||
the seed is the randomness of a given seed compared to the
|
||
previous seed used, and this affects the randomness of separate
|
||
number sequences. The method for choosing a seed is critical if
|
||
the generated numbers are to be used for important applications,
|
||
such as generating cryptographic keys.
|
||
|
||
Traditionally the system time has been used to seed, but care
|
||
needs to be taken with this. If an application seeds often and
|
||
the resolution of the system clock is low, then the same sequence
|
||
of numbers might be repeated. Also, the system time is quite easy
|
||
to guess, so if unpredictability is required then it should
|
||
definitely not be the only source for the seed value. On some
|
||
systems there's a special device `/dev/random' which provides
|
||
random data better suited for use as a seed.
|
||
|
||
|
||
File: mpir.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions
|
||
|
||
9.3 Random State Miscellaneous
|
||
==============================
|
||
|
||
-- Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE,
|
||
unsigned long N)
|
||
Return a uniformly distributed random number of N bits, ie. in the
|
||
range 0 to 2^N-1 inclusive. N must be less than or equal to the
|
||
number of bits in an `unsigned long'.
|
||
|
||
-- Function: unsigned long gmp_urandomm_ui (gmp_randstate_t STATE,
|
||
unsigned long N)
|
||
Return a uniformly distributed random number in the range 0 to
|
||
N-1, inclusive.
|
||
|
||
|
||
File: mpir.info, Node: Formatted Output, Next: Formatted Input, Prev: Random Number Functions, Up: Top
|
||
|
||
10 Formatted Output
|
||
*******************
|
||
|
||
* Menu:
|
||
|
||
* Formatted Output Strings::
|
||
* Formatted Output Functions::
|
||
* C++ Formatted Output::
|
||
|
||
|
||
File: mpir.info, Node: Formatted Output Strings, Next: Formatted Output Functions, Prev: Formatted Output, Up: Formatted Output
|
||
|
||
10.1 Format Strings
|
||
===================
|
||
|
||
`gmp_printf' and friends accept format strings similar to the standard C
|
||
`printf' (*note Formatted Output: (libc)Formatted Output.). A format
|
||
specification is of the form
|
||
|
||
% [flags] [width] [.[precision]] [type] conv
|
||
|
||
MPIR adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
|
||
respectively, `M' for `mp_limb_t', and `N' for an `mp_limb_t' array.
|
||
`Z', `Q', `M' and `N' behave like integers. `Q' will print a `/' and a
|
||
denominator, if needed. `F' behaves like a float. For example,
|
||
|
||
mpz_t z;
|
||
gmp_printf ("%s is an mpz %Zd\n", "here", z);
|
||
|
||
mpq_t q;
|
||
gmp_printf ("a hex rational: %#40Qx\n", q);
|
||
|
||
mpf_t f;
|
||
int n;
|
||
gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
|
||
|
||
mp_limb_t l;
|
||
gmp_printf ("limb %Mu\n", limb);
|
||
|
||
const mp_limb_t *ptr;
|
||
mp_size_t size;
|
||
gmp_printf ("limb array %Nx\n", ptr, size);
|
||
|
||
For `N' the limbs are expected least significant first, as per the
|
||
`mpn' functions (*note Low-level Functions::). A negative size can be
|
||
given to print the value as a negative.
|
||
|
||
All the standard C `printf' types behave the same as the C library
|
||
`printf', and can be freely intermixed with the MPIR extensions. In the
|
||
current implementation the standard parts of the format string are
|
||
simply handed to `printf' and only the MPIR extensions handled directly.
|
||
|
||
The flags accepted are as follows. GLIBC style ' is only for the
|
||
standard C types (not the MPIR types), and only if the C library
|
||
supports it.
|
||
|
||
0 pad with zeros (rather than spaces)
|
||
# show the base with `0x', `0X' or `0'
|
||
+ always show a sign
|
||
(space) show a space or a `-' sign
|
||
' group digits, GLIBC style (not MPIR
|
||
types)
|
||
|
||
The optional width and precision can be given as a number within the
|
||
format string, or as a `*' to take an extra parameter of type `int', the
|
||
same as the standard `printf'.
|
||
|
||
The standard types accepted are as follows. `h' and `l' are
|
||
portable, the rest will depend on the compiler (or include files) for
|
||
the type and the C library for the output.
|
||
|
||
h short
|
||
hh char
|
||
j intmax_t or uintmax_t
|
||
l long or wchar_t
|
||
ll long long
|
||
L long double
|
||
q quad_t or u_quad_t
|
||
t ptrdiff_t
|
||
z size_t
|
||
|
||
The MPIR types are
|
||
|
||
F mpf_t, float conversions
|
||
Q mpq_t, integer conversions
|
||
M mp_limb_t, integer conversions
|
||
N mp_limb_t array, integer conversions
|
||
Z mpz_t, integer conversions
|
||
|
||
The conversions accepted are as follows. `a' and `A' are always
|
||
supported for `mpf_t' but depend on the C library for standard C float
|
||
types. `m' and `p' depend on the C library.
|
||
|
||
a A hex floats, C99 style
|
||
c character
|
||
d decimal integer
|
||
e E scientific format float
|
||
f fixed point float
|
||
i same as d
|
||
g G fixed or scientific float
|
||
m `strerror' string, GLIBC style
|
||
n store characters written so far
|
||
o octal integer
|
||
p pointer
|
||
s string
|
||
u unsigned integer
|
||
x X hex integer
|
||
|
||
`o', `x' and `X' are unsigned for the standard C types, but for
|
||
types `Z', `Q' and `N' they are signed. `u' is not meaningful for `Z',
|
||
`Q' and `N'.
|
||
|
||
`M' is a proxy for the C library `l' or `L', according to the size
|
||
of `mp_limb_t'. Unsigned conversions will be usual, but a signed
|
||
conversion can be used and will interpret the value as a twos complement
|
||
negative.
|
||
|
||
`n' can be used with any type, even the MPIR types.
|
||
|
||
Other types or conversions that might be accepted by the C library
|
||
`printf' cannot be used through `gmp_printf', this includes for
|
||
instance extensions registered with GLIBC `register_printf_function'.
|
||
Also currently there's no support for POSIX `$' style numbered arguments
|
||
(perhaps this will be added in the future).
|
||
|
||
The precision field has it's usual meaning for integer `Z' and float
|
||
`F' types, but is currently undefined for `Q' and should not be used
|
||
with that.
|
||
|
||
`mpf_t' conversions only ever generate as many digits as can be
|
||
accurately represented by the operand, the same as `mpf_get_str' does.
|
||
Zeros will be used if necessary to pad to the requested precision. This
|
||
happens even for an `f' conversion of an `mpf_t' which is an integer,
|
||
for instance 2^1024 in an `mpf_t' of 128 bits precision will only
|
||
produce about 40 digits, then pad with zeros to the decimal point. An
|
||
empty precision field like `%.Fe' or `%.Ff' can be used to specifically
|
||
request just the significant digits.
|
||
|
||
The decimal point character (or string) is taken from the current
|
||
locale settings on systems which provide `localeconv' (*note Locales
|
||
and Internationalization: (libc)Locales.). The C library will normally
|
||
do the same for standard float output.
|
||
|
||
The format string is only interpreted as plain `char's, multibyte
|
||
characters are not recognised. Perhaps this will change in the future.
|
||
|
||
|
||
File: mpir.info, Node: Formatted Output Functions, Next: C++ Formatted Output, Prev: Formatted Output Strings, Up: Formatted Output
|
||
|
||
10.2 Functions
|
||
==============
|
||
|
||
Each of the following functions is similar to the corresponding C
|
||
library function. The basic `printf' forms take a variable argument
|
||
list. The `vprintf' forms take an argument pointer, see *note Variadic
|
||
Functions: (libc)Variadic Functions, or `man 3 va_start'.
|
||
|
||
It should be emphasised that if a format string is invalid, or the
|
||
arguments don't match what the format specifies, then the behaviour of
|
||
any of these functions will be unpredictable. GCC format string
|
||
checking is not available, since it doesn't recognise the MPIR
|
||
extensions.
|
||
|
||
The file based functions `gmp_printf' and `gmp_fprintf' will return
|
||
-1 to indicate a write error. Output is not "atomic", so partial
|
||
output may be produced if a write error occurs. All the functions can
|
||
return -1 if the C library `printf' variant in use returns -1, but this
|
||
shouldn't normally occur.
|
||
|
||
-- Function: int gmp_printf (const char *FMT, ...)
|
||
-- Function: int gmp_vprintf (const char *FMT, va_list AP)
|
||
Print to the standard output `stdout'. Return the number of
|
||
characters written, or -1 if an error occurred.
|
||
|
||
-- Function: int gmp_fprintf (FILE *FP, const char *FMT, ...)
|
||
-- Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP)
|
||
Print to the stream FP. Return the number of characters written,
|
||
or -1 if an error occurred.
|
||
|
||
-- Function: int gmp_sprintf (char *BUF, const char *FMT, ...)
|
||
-- Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP)
|
||
Form a null-terminated string in BUF. Return the number of
|
||
characters written, excluding the terminating null.
|
||
|
||
No overlap is permitted between the space at BUF and the string
|
||
FMT.
|
||
|
||
These functions are not recommended, since there's no protection
|
||
against exceeding the space available at BUF.
|
||
|
||
-- Function: int gmp_snprintf (char *BUF, size_t SIZE, const char
|
||
*FMT, ...)
|
||
-- Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char
|
||
*FMT, va_list AP)
|
||
Form a null-terminated string in BUF. No more than SIZE bytes
|
||
will be written. To get the full output, SIZE must be enough for
|
||
the string and null-terminator.
|
||
|
||
The return value is the total number of characters which ought to
|
||
have been produced, excluding the terminating null. If RETVAL >=
|
||
SIZE then the actual output has been truncated to the first SIZE-1
|
||
characters, and a null appended.
|
||
|
||
No overlap is permitted between the region {BUF,SIZE} and the FMT
|
||
string.
|
||
|
||
Notice the return value is in ISO C99 `snprintf' style. This is
|
||
so even if the C library `vsnprintf' is the older GLIBC 2.0.x
|
||
style.
|
||
|
||
-- Function: int gmp_asprintf (char **PP, const char *FMT, ...)
|
||
-- Function: int gmp_vasprintf (char **PP, const char *FMT, va_list AP)
|
||
Form a null-terminated string in a block of memory obtained from
|
||
the current memory allocation function (*note Custom
|
||
Allocation::). The block will be the size of the string and
|
||
null-terminator. The address of the block in stored to *PP. The
|
||
return value is the number of characters produced, excluding the
|
||
null-terminator.
|
||
|
||
Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1
|
||
if there's no more memory available, it lets the current allocation
|
||
function handle that.
|
||
|
||
-- Function: int gmp_obstack_printf (struct obstack *OB, const char
|
||
*FMT, ...)
|
||
-- Function: int gmp_obstack_vprintf (struct obstack *OB, const char
|
||
*FMT, va_list AP)
|
||
Append to the current object in OB. The return value is the
|
||
number of characters written. A null-terminator is not written.
|
||
|
||
FMT cannot be within the current object in OB, since that object
|
||
might move as it grows.
|
||
|
||
These functions are available only when the C library provides the
|
||
obstack feature, which probably means only on GNU systems, see
|
||
*note Obstacks: (libc)Obstacks.
|
||
|
||
|
||
File: mpir.info, Node: C++ Formatted Output, Prev: Formatted Output Functions, Up: Formatted Output
|
||
|
||
10.3 C++ Formatted Output
|
||
=========================
|
||
|
||
The following functions are provided in `libmpirxx' (*note Headers and
|
||
Libraries::), which is built if C++ support is enabled (*note Build
|
||
Options::). Prototypes are available from `<mpir.h>'.
|
||
|
||
-- Function: ostream& operator<< (ostream& STREAM, mpz_t OP)
|
||
Print OP to STREAM, using its `ios' formatting settings.
|
||
`ios::width' is reset to 0 after output, the same as the standard
|
||
`ostream operator<<' routines do.
|
||
|
||
In hex or octal, OP is printed as a signed number, the same as for
|
||
decimal. This is unlike the standard `operator<<' routines on
|
||
`int' etc, which instead give twos complement.
|
||
|
||
-- Function: ostream& operator<< (ostream& STREAM, mpq_t OP)
|
||
Print OP to STREAM, using its `ios' formatting settings.
|
||
`ios::width' is reset to 0 after output, the same as the standard
|
||
`ostream operator<<' routines do.
|
||
|
||
Output will be a fraction like `5/9', or if the denominator is 1
|
||
then just a plain integer like `123'.
|
||
|
||
In hex or octal, OP is printed as a signed value, the same as for
|
||
decimal. If `ios::showbase' is set then a base indicator is shown
|
||
on both the numerator and denominator (if the denominator is
|
||
required).
|
||
|
||
-- Function: ostream& operator<< (ostream& STREAM, mpf_t OP)
|
||
Print OP to STREAM, using its `ios' formatting settings.
|
||
`ios::width' is reset to 0 after output, the same as the standard
|
||
`ostream operator<<' routines do.
|
||
|
||
The decimal point follows the standard library float `operator<<',
|
||
which on recent systems means the `std::locale' imbued on STREAM.
|
||
|
||
Hex and octal are supported, unlike the standard `operator<<' on
|
||
`double'. The mantissa will be in hex or octal, the exponent will
|
||
be in decimal. For hex the exponent delimiter is an `@'. This is
|
||
as per `mpf_out_str'.
|
||
|
||
`ios::showbase' is supported, and will put a base on the mantissa,
|
||
for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'.
|
||
This last form is slightly strange, but at least differentiates
|
||
itself from decimal.
|
||
|
||
These operators mean that MPIR types can be printed in the usual C++
|
||
way, for example,
|
||
|
||
mpz_t z;
|
||
int n;
|
||
...
|
||
cout << "iteration " << n << " value " << z << "\n";
|
||
|
||
But note that `ostream' output (and `istream' input, *note C++
|
||
Formatted Input::) is the only overloading available for the MPIR types
|
||
and that for instance using `+' with an `mpz_t' will have unpredictable
|
||
results. For classes with overloading, see *note C++ Class Interface::.
|
||
|
||
|
||
File: mpir.info, Node: Formatted Input, Next: C++ Class Interface, Prev: Formatted Output, Up: Top
|
||
|
||
11 Formatted Input
|
||
******************
|
||
|
||
* Menu:
|
||
|
||
* Formatted Input Strings::
|
||
* Formatted Input Functions::
|
||
* C++ Formatted Input::
|
||
|
||
|
||
File: mpir.info, Node: Formatted Input Strings, Next: Formatted Input Functions, Prev: Formatted Input, Up: Formatted Input
|
||
|
||
11.1 Formatted Input Strings
|
||
============================
|
||
|
||
`gmp_scanf' and friends accept format strings similar to the standard C
|
||
`scanf' (*note Formatted Input: (libc)Formatted Input.). A format
|
||
specification is of the form
|
||
|
||
% [flags] [width] [type] conv
|
||
|
||
MPIR adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
|
||
respectively. `Z' and `Q' behave like integers. `Q' will read a `/'
|
||
and a denominator, if present. `F' behaves like a float.
|
||
|
||
MPIR variables don't require an `&' when passed to `gmp_scanf', since
|
||
they're already "call-by-reference". For example,
|
||
|
||
/* to read say "a(5) = 1234" */
|
||
int n;
|
||
mpz_t z;
|
||
gmp_scanf ("a(%d) = %Zd\n", &n, z);
|
||
|
||
mpq_t q1, q2;
|
||
gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
|
||
|
||
/* to read say "topleft (1.55,-2.66)" */
|
||
mpf_t x, y;
|
||
char buf[32];
|
||
gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
|
||
|
||
All the standard C `scanf' types behave the same as in the C library
|
||
`scanf', and can be freely intermixed with the MPIR extensions. In the
|
||
current implementation the standard parts of the format string are
|
||
simply handed to `scanf' and only the MPIR extensions handled directly.
|
||
|
||
The flags accepted are as follows. `a' and `'' will depend on
|
||
support from the C library, and `'' cannot be used with MPIR types.
|
||
|
||
* read but don't store
|
||
a allocate a buffer (string conversions)
|
||
' grouped digits, GLIBC style (not MPIR
|
||
types)
|
||
|
||
The standard types accepted are as follows. `h' and `l' are
|
||
portable, the rest will depend on the compiler (or include files) for
|
||
the type and the C library for the input.
|
||
|
||
h short
|
||
hh char
|
||
j intmax_t or uintmax_t
|
||
l long int, double or wchar_t
|
||
ll long long
|
||
L long double
|
||
q quad_t or u_quad_t
|
||
t ptrdiff_t
|
||
z size_t
|
||
|
||
The MPIR types are
|
||
|
||
F mpf_t, float conversions
|
||
Q mpq_t, integer conversions
|
||
Z mpz_t, integer conversions
|
||
|
||
The conversions accepted are as follows. `p' and `[' will depend on
|
||
support from the C library, the rest are standard.
|
||
|
||
c character or characters
|
||
d decimal integer
|
||
e E f g G float
|
||
i integer with base indicator
|
||
n characters read so far
|
||
o octal integer
|
||
p pointer
|
||
s string of non-whitespace characters
|
||
u decimal integer
|
||
x X hex integer
|
||
[ string of characters in a set
|
||
|
||
`e', `E', `f', `g' and `G' are identical, they all read either fixed
|
||
point or scientific format, and either upper or lower case `e' for the
|
||
exponent in scientific format.
|
||
|
||
C99 style hex float format (`printf %a', *note Formatted Output
|
||
Strings::) is always accepted for `mpf_t', but for the standard float
|
||
types it will depend on the C library.
|
||
|
||
`x' and `X' are identical, both accept both upper and lower case
|
||
hexadecimal.
|
||
|
||
`o', `u', `x' and `X' all read positive or negative values. For the
|
||
standard C types these are described as "unsigned" conversions, but
|
||
that merely affects certain overflow handling, negatives are still
|
||
allowed (per `strtoul', *note Parsing of Integers: (libc)Parsing of
|
||
Integers.). For MPIR types there are no overflows, so `d' and `u' are
|
||
identical.
|
||
|
||
`Q' type reads the numerator and (optional) denominator as given.
|
||
If the value might not be in canonical form then `mpq_canonicalize'
|
||
must be called before using it in any calculations (*note Rational
|
||
Number Functions::).
|
||
|
||
`Qi' will read a base specification separately for the numerator and
|
||
denominator. For example `0x10/11' would be 16/11, whereas `0x10/0x11'
|
||
would be 16/17.
|
||
|
||
`n' can be used with any of the types above, even the MPIR types.
|
||
`*' to suppress assignment is allowed, though in that case it would do
|
||
nothing at all.
|
||
|
||
Other conversions or types that might be accepted by the C library
|
||
`scanf' cannot be used through `gmp_scanf'.
|
||
|
||
Whitespace is read and discarded before a field, except for `c' and
|
||
`[' conversions.
|
||
|
||
For float conversions, the decimal point character (or string)
|
||
expected is taken from the current locale settings on systems which
|
||
provide `localeconv' (*note Locales and Internationalization:
|
||
(libc)Locales.). The C library will normally do the same for standard
|
||
float input.
|
||
|
||
The format string is only interpreted as plain `char's, multibyte
|
||
characters are not recognised. Perhaps this will change in the future.
|
||
|
||
|
||
File: mpir.info, Node: Formatted Input Functions, Next: C++ Formatted Input, Prev: Formatted Input Strings, Up: Formatted Input
|
||
|
||
11.2 Formatted Input Functions
|
||
==============================
|
||
|
||
Each of the following functions is similar to the corresponding C
|
||
library function. The plain `scanf' forms take a variable argument
|
||
list. The `vscanf' forms take an argument pointer, see *note Variadic
|
||
Functions: (libc)Variadic Functions, or `man 3 va_start'.
|
||
|
||
It should be emphasised that if a format string is invalid, or the
|
||
arguments don't match what the format specifies, then the behaviour of
|
||
any of these functions will be unpredictable. GCC format string
|
||
checking is not available, since it doesn't recognise the MPIR
|
||
extensions.
|
||
|
||
No overlap is permitted between the FMT string and any of the results
|
||
produced.
|
||
|
||
-- Function: int gmp_scanf (const char *FMT, ...)
|
||
-- Function: int gmp_vscanf (const char *FMT, va_list AP)
|
||
Read from the standard input `stdin'.
|
||
|
||
-- Function: int gmp_fscanf (FILE *FP, const char *FMT, ...)
|
||
-- Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP)
|
||
Read from the stream FP.
|
||
|
||
-- Function: int gmp_sscanf (const char *S, const char *FMT, ...)
|
||
-- Function: int gmp_vsscanf (const char *S, const char *FMT, va_list
|
||
AP)
|
||
Read from a null-terminated string S.
|
||
|
||
The return value from each of these functions is the same as the
|
||
standard C99 `scanf', namely the number of fields successfully parsed
|
||
and stored. `%n' fields and fields read but suppressed by `*' don't
|
||
count towards the return value.
|
||
|
||
If end of input (or a file error) is reached before a character for
|
||
a field or a literal, and if no previous non-suppressed fields have
|
||
matched, then the return value is `EOF' instead of 0. A whitespace
|
||
character in the format string is only an optional match and doesn't
|
||
induce an `EOF' in this fashion. Leading whitespace read and discarded
|
||
for a field don't count as characters for that field.
|
||
|
||
For the MPIR types, input parsing follows C99 rules, namely one
|
||
character of lookahead is used and characters are read while they
|
||
continue to meet the format requirements. If this doesn't provide a
|
||
complete number then the function terminates, with that field not
|
||
stored nor counted towards the return value. For instance with `mpf_t'
|
||
an input `1.23e-XYZ' would be read up to the `X' and that character
|
||
pushed back since it's not a digit. The string `1.23e-' would then be
|
||
considered invalid since an `e' must be followed by at least one digit.
|
||
|
||
For the standard C types, in the current implementation MPIR calls
|
||
the C library `scanf' functions, which might have looser rules about
|
||
what constitutes a valid input.
|
||
|
||
Note that `gmp_sscanf' is the same as `gmp_fscanf' and only does one
|
||
character of lookahead when parsing. Although clearly it could look at
|
||
its entire input, it is deliberately made identical to `gmp_fscanf',
|
||
the same way C99 `sscanf' is the same as `fscanf'.
|
||
|
||
|
||
File: mpir.info, Node: C++ Formatted Input, Prev: Formatted Input Functions, Up: Formatted Input
|
||
|
||
11.3 C++ Formatted Input
|
||
========================
|
||
|
||
The following functions are provided in `libmpirxx' (*note Headers and
|
||
Libraries::), which is built only if C++ support is enabled (*note
|
||
Build Options::). Prototypes are available from `<mpir.h>'.
|
||
|
||
-- Function: istream& operator>> (istream& STREAM, mpz_t ROP)
|
||
Read ROP from STREAM, using its `ios' formatting settings.
|
||
|
||
-- Function: istream& operator>> (istream& STREAM, mpq_t ROP)
|
||
An integer like `123' will be read, or a fraction like `5/9'. No
|
||
whitespace is allowed around the `/'. If the fraction is not in
|
||
canonical form then `mpq_canonicalize' must be called (*note
|
||
Rational Number Functions::) before operating on it.
|
||
|
||
As per integer input, an `0' or `0x' base indicator is read when
|
||
none of `ios::dec', `ios::oct' or `ios::hex' are set. This is
|
||
done separately for numerator and denominator, so that for instance
|
||
`0x10/11' is 16/11 and `0x10/0x11' is 16/17.
|
||
|
||
-- Function: istream& operator>> (istream& STREAM, mpf_t ROP)
|
||
Read ROP from STREAM, using its `ios' formatting settings.
|
||
|
||
Hex or octal floats are not supported, but might be in the future,
|
||
or perhaps it's best to accept only what the standard float
|
||
`operator>>' does.
|
||
|
||
Note that digit grouping specified by the `istream' locale is
|
||
currently not accepted. Perhaps this will change in the future.
|
||
|
||
|
||
These operators mean that MPIR types can be read in the usual C++
|
||
way, for example,
|
||
|
||
mpz_t z;
|
||
...
|
||
cin >> z;
|
||
|
||
But note that `istream' input (and `ostream' output, *note C++
|
||
Formatted Output::) is the only overloading available for the MPIR
|
||
types and that for instance using `+' with an `mpz_t' will have
|
||
unpredictable results. For classes with overloading, see *note C++
|
||
Class Interface::.
|
||
|
||
|
||
File: mpir.info, Node: C++ Class Interface, Next: BSD Compatible Functions, Prev: Formatted Input, Up: Top
|
||
|
||
12 C++ Class Interface
|
||
**********************
|
||
|
||
This chapter describes the C++ class based interface to MPIR.
|
||
|
||
All MPIR C language types and functions can be used in C++ programs,
|
||
since `mpir.h' has `extern "C"' qualifiers, but the class interface
|
||
offers overloaded functions and operators which may be more convenient.
|
||
|
||
Due to the implementation of this interface, a reasonably recent C++
|
||
compiler is required, one supporting namespaces, partial specialization
|
||
of templates and member templates. For GCC this means version 2.91 or
|
||
later.
|
||
|
||
*Everything described in this chapter is to be considered preliminary
|
||
and might be subject to incompatible changes if some unforeseen
|
||
difficulty reveals itself.*
|
||
|
||
* Menu:
|
||
|
||
* C++ Interface General::
|
||
* C++ Interface Integers::
|
||
* C++ Interface Rationals::
|
||
* C++ Interface Floats::
|
||
* C++ Interface Random Numbers::
|
||
* C++ Interface Limitations::
|
||
|
||
|
||
File: mpir.info, Node: C++ Interface General, Next: C++ Interface Integers, Prev: C++ Class Interface, Up: C++ Class Interface
|
||
|
||
12.1 C++ Interface General
|
||
==========================
|
||
|
||
All the C++ classes and functions are available with
|
||
|
||
#include <mpirxx.h>
|
||
|
||
Programs should be linked with the `libmpirxx' and `libmpir'
|
||
libraries. For example,
|
||
|
||
g++ mycxxprog.cc -lmpirxx -lmpir
|
||
|
||
The classes defined are
|
||
|
||
-- Class: mpz_class
|
||
-- Class: mpq_class
|
||
-- Class: mpf_class
|
||
|
||
The standard operators and various standard functions are overloaded
|
||
to allow arithmetic with these classes. For example,
|
||
|
||
int
|
||
main (void)
|
||
{
|
||
mpz_class a, b, c;
|
||
|
||
a = 1234;
|
||
b = "-5678";
|
||
c = a+b;
|
||
cout << "sum is " << c << "\n";
|
||
cout << "absolute value is " << abs(c) << "\n";
|
||
|
||
return 0;
|
||
}
|
||
|
||
An important feature of the implementation is that an expression like
|
||
`a=b+c' results in a single call to the corresponding `mpz_add',
|
||
without using a temporary for the `b+c' part. Expressions which by
|
||
their nature imply intermediate values, like `a=b*c+d*e', still use
|
||
temporaries though.
|
||
|
||
The classes can be freely intermixed in expressions, as can the
|
||
classes and the standard types `long', `unsigned long' and `double'.
|
||
Smaller types like `int' or `float' can also be intermixed, since C++
|
||
will promote them.
|
||
|
||
Note that `bool' is not accepted directly, but must be explicitly
|
||
cast to an `int' first. This is because C++ will automatically convert
|
||
any pointer to a `bool', so if MPIR accepted `bool' it would make all
|
||
sorts of invalid class and pointer combinations compile but almost
|
||
certainly not do anything sensible.
|
||
|
||
Conversions back from the classes to standard C++ types aren't done
|
||
automatically, instead member functions like `get_si' are provided (see
|
||
the following sections for details).
|
||
|
||
Also there are no automatic conversions from the classes to the
|
||
corresponding MPIR C types, instead a reference to the underlying C
|
||
object can be obtained with the following functions,
|
||
|
||
-- Function: mpz_t mpz_class::get_mpz_t ()
|
||
-- Function: mpq_t mpq_class::get_mpq_t ()
|
||
-- Function: mpf_t mpf_class::get_mpf_t ()
|
||
|
||
These can be used to call a C function which doesn't have a C++ class
|
||
interface. For example to set `a' to the GCD of `b' and `c',
|
||
|
||
mpz_class a, b, c;
|
||
...
|
||
mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
|
||
|
||
In the other direction, a class can be initialized from the
|
||
corresponding MPIR C type, or assigned to if an explicit constructor is
|
||
used. In both cases this makes a copy of the value, it doesn't create
|
||
any sort of association. For example,
|
||
|
||
mpz_t z;
|
||
// ... init and calculate z ...
|
||
mpz_class x(z);
|
||
mpz_class y;
|
||
y = mpz_class (z);
|
||
|
||
There are no namespace setups in `mpirxx.h', all types and functions
|
||
are simply put into the global namespace. This is what `mpir.h' has
|
||
done in the past, and continues to do for compatibility. The extras
|
||
provided by `mpirxx.h' follow MPIR naming conventions and are unlikely
|
||
to clash with anything.
|
||
|
||
|
||
File: mpir.info, Node: C++ Interface Integers, Next: C++ Interface Rationals, Prev: C++ Interface General, Up: C++ Class Interface
|
||
|
||
12.2 C++ Interface Integers
|
||
===========================
|
||
|
||
-- Function: void mpz_class::mpz_class (type N)
|
||
Construct an `mpz_class'. All the standard C++ types may be used,
|
||
except `long long' and `long double', and all the MPIR C++ classes
|
||
can be used. Any necessary conversion follows the corresponding C
|
||
function, for example `double' follows `mpz_set_d' (*note
|
||
Assigning Integers::).
|
||
|
||
-- Function: void mpz_class::mpz_class (mpz_t Z)
|
||
Construct an `mpz_class' from an `mpz_t'. The value in Z is
|
||
copied into the new `mpz_class', there won't be any permanent
|
||
association between it and Z.
|
||
|
||
-- Function: void mpz_class::mpz_class (const char *S)
|
||
-- Function: void mpz_class::mpz_class (const char *S, int BASE = 0)
|
||
-- Function: void mpz_class::mpz_class (const string& S)
|
||
-- Function: void mpz_class::mpz_class (const string& S, int BASE = 0)
|
||
Construct an `mpz_class' converted from a string using
|
||
`mpz_set_str' (*note Assigning Integers::).
|
||
|
||
If the string is not a valid integer, an `std::invalid_argument'
|
||
exception is thrown. The same applies to `operator='.
|
||
|
||
-- Function: mpz_class operator/ (mpz_class A, mpz_class D)
|
||
-- Function: mpz_class operator% (mpz_class A, mpz_class D)
|
||
Divisions involving `mpz_class' round towards zero, as per the
|
||
`mpz_tdiv_q' and `mpz_tdiv_r' functions (*note Integer Division::).
|
||
This is the same as the C99 `/' and `%' operators.
|
||
|
||
The `mpz_fdiv...' or `mpz_cdiv...' functions can always be called
|
||
directly if desired. For example,
|
||
|
||
mpz_class q, a, d;
|
||
...
|
||
mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
|
||
|
||
-- Function: mpz_class abs (mpz_class OP1)
|
||
-- Function: int cmp (mpz_class OP1, type OP2)
|
||
-- Function: int cmp (type OP1, mpz_class OP2)
|
||
-- Function: bool mpz_class::fits_sint_p (void)
|
||
-- Function: bool mpz_class::fits_slong_p (void)
|
||
-- Function: bool mpz_class::fits_sshort_p (void)
|
||
-- Function: bool mpz_class::fits_uint_p (void)
|
||
-- Function: bool mpz_class::fits_ulong_p (void)
|
||
-- Function: bool mpz_class::fits_ushort_p (void)
|
||
-- Function: double mpz_class::get_d (void)
|
||
-- Function: long mpz_class::get_si (void)
|
||
-- Function: string mpz_class::get_str (int BASE = 10)
|
||
-- Function: unsigned long mpz_class::get_ui (void)
|
||
-- Function: int mpz_class::set_str (const char *STR, int BASE)
|
||
-- Function: int mpz_class::set_str (const string& STR, int BASE)
|
||
-- Function: int sgn (mpz_class OP)
|
||
-- Function: mpz_class sqrt (mpz_class OP)
|
||
These functions provide a C++ class interface to the corresponding
|
||
MPIR C routines.
|
||
|
||
`cmp' can be used with any of the classes or the standard C++
|
||
types, except `long long' and `long double'.
|
||
|
||
|
||
Overloaded operators for combinations of `mpz_class' and `double'
|
||
are provided for completeness, but it should be noted that if the given
|
||
`double' is not an integer then the way any rounding is done is
|
||
currently unspecified. The rounding might take place at the start, in
|
||
the middle, or at the end of the operation, and it might change in the
|
||
future.
|
||
|
||
Conversions between `mpz_class' and `double', however, are defined
|
||
to follow the corresponding C functions `mpz_get_d' and `mpz_set_d'.
|
||
And comparisons are always made exactly, as per `mpz_cmp_d'.
|
||
|
||
|
||
File: mpir.info, Node: C++ Interface Rationals, Next: C++ Interface Floats, Prev: C++ Interface Integers, Up: C++ Class Interface
|
||
|
||
12.3 C++ Interface Rationals
|
||
============================
|
||
|
||
In all the following constructors, if a fraction is given then it
|
||
should be in canonical form, or if not then `mpq_class::canonicalize'
|
||
called.
|
||
|
||
-- Function: void mpq_class::mpq_class (type OP)
|
||
-- Function: void mpq_class::mpq_class (integer NUM, integer DEN)
|
||
Construct an `mpq_class'. The initial value can be a single value
|
||
of any type, or a pair of integers (`mpz_class' or standard C++
|
||
integer types) representing a fraction, except that `long long'
|
||
and `long double' are not supported. For example,
|
||
|
||
mpq_class q (99);
|
||
mpq_class q (1.75);
|
||
mpq_class q (1, 3);
|
||
|
||
-- Function: void mpq_class::mpq_class (mpq_t Q)
|
||
Construct an `mpq_class' from an `mpq_t'. The value in Q is
|
||
copied into the new `mpq_class', there won't be any permanent
|
||
association between it and Q.
|
||
|
||
-- Function: void mpq_class::mpq_class (const char *S)
|
||
-- Function: void mpq_class::mpq_class (const char *S, int BASE = 0)
|
||
-- Function: void mpq_class::mpq_class (const string& S)
|
||
-- Function: void mpq_class::mpq_class (const string& S, int BASE = 0)
|
||
Construct an `mpq_class' converted from a string using
|
||
`mpq_set_str' (*note Initializing Rationals::).
|
||
|
||
If the string is not a valid rational, an `std::invalid_argument'
|
||
exception is thrown. The same applies to `operator='.
|
||
|
||
-- Function: void mpq_class::canonicalize ()
|
||
Put an `mpq_class' into canonical form, as per *note Rational
|
||
Number Functions::. All arithmetic operators require their
|
||
operands in canonical form, and will return results in canonical
|
||
form.
|
||
|
||
-- Function: mpq_class abs (mpq_class OP)
|
||
-- Function: int cmp (mpq_class OP1, type OP2)
|
||
-- Function: int cmp (type OP1, mpq_class OP2)
|
||
-- Function: double mpq_class::get_d (void)
|
||
-- Function: string mpq_class::get_str (int BASE = 10)
|
||
-- Function: int mpq_class::set_str (const char *STR, int BASE)
|
||
-- Function: int mpq_class::set_str (const string& STR, int BASE)
|
||
-- Function: int sgn (mpq_class OP)
|
||
These functions provide a C++ class interface to the corresponding
|
||
MPIR C routines.
|
||
|
||
`cmp' can be used with any of the classes or the standard C++
|
||
types, except `long long' and `long double'.
|
||
|
||
-- Function: mpz_class& mpq_class::get_num ()
|
||
-- Function: mpz_class& mpq_class::get_den ()
|
||
Get a reference to an `mpz_class' which is the numerator or
|
||
denominator of an `mpq_class'. This can be used both for read and
|
||
write access. If the object returned is modified, it modifies the
|
||
original `mpq_class'.
|
||
|
||
If direct manipulation might produce a non-canonical value, then
|
||
`mpq_class::canonicalize' must be called before further operations.
|
||
|
||
-- Function: mpz_t mpq_class::get_num_mpz_t ()
|
||
-- Function: mpz_t mpq_class::get_den_mpz_t ()
|
||
Get a reference to the underlying `mpz_t' numerator or denominator
|
||
of an `mpq_class'. This can be passed to C functions expecting an
|
||
`mpz_t'. Any modifications made to the `mpz_t' will modify the
|
||
original `mpq_class'.
|
||
|
||
If direct manipulation might produce a non-canonical value, then
|
||
`mpq_class::canonicalize' must be called before further operations.
|
||
|
||
-- Function: istream& operator>> (istream& STREAM, mpq_class& ROP);
|
||
Read ROP from STREAM, using its `ios' formatting settings, the
|
||
same as `mpq_t operator>>' (*note C++ Formatted Input::).
|
||
|
||
If the ROP read might not be in canonical form then
|
||
`mpq_class::canonicalize' must be called.
|
||
|
||
|
||
File: mpir.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface
|
||
|
||
12.4 C++ Interface Floats
|
||
=========================
|
||
|
||
When an expression requires the use of temporary intermediate
|
||
`mpf_class' values, like `f=g*h+x*y', those temporaries will have the
|
||
same precision as the destination `f'. Explicit constructors can be
|
||
used if this doesn't suit.
|
||
|
||
-- Function: mpf_class::mpf_class (type OP)
|
||
-- Function: mpf_class::mpf_class (type OP, unsigned long PREC)
|
||
Construct an `mpf_class'. Any standard C++ type can be used,
|
||
except `long long' and `long double', and any of the MPIR C++
|
||
classes can be used.
|
||
|
||
If PREC is given, the initial precision is that value, in bits. If
|
||
PREC is not given, then the initial precision is determined by the
|
||
type of OP given. An `mpz_class', `mpq_class', or C++ builtin
|
||
type will give the default `mpf' precision (*note Initializing
|
||
Floats::). An `mpf_class' or expression will give the precision
|
||
of that value. The precision of a binary expression is the higher
|
||
of the two operands.
|
||
|
||
mpf_class f(1.5); // default precision
|
||
mpf_class f(1.5, 500); // 500 bits (at least)
|
||
mpf_class f(x); // precision of x
|
||
mpf_class f(abs(x)); // precision of x
|
||
mpf_class f(-g, 1000); // 1000 bits (at least)
|
||
mpf_class f(x+y); // greater of precisions of x and y
|
||
|
||
-- Function: void mpf_class::mpf_class (const char *S)
|
||
-- Function: void mpf_class::mpf_class (const char *S, unsigned long
|
||
PREC, int BASE = 0)
|
||
-- Function: void mpf_class::mpf_class (const string& S)
|
||
-- Function: void mpf_class::mpf_class (const string& S, unsigned long
|
||
PREC, int BASE = 0)
|
||
Construct an `mpf_class' converted from a string using
|
||
`mpf_set_str' (*note Assigning Floats::). If PREC is given, the
|
||
initial precision is that value, in bits. If not, the default
|
||
`mpf' precision (*note Initializing Floats::) is used.
|
||
|
||
If the string is not a valid float, an `std::invalid_argument'
|
||
exception is thrown. The same applies to `operator='.
|
||
|
||
-- Function: mpf_class& mpf_class::operator= (type OP)
|
||
Convert and store the given OP value to an `mpf_class' object. The
|
||
same types are accepted as for the constructors above.
|
||
|
||
Note that `operator=' only stores a new value, it doesn't copy or
|
||
change the precision of the destination, instead the value is
|
||
truncated if necessary. This is the same as `mpf_set' etc. Note
|
||
in particular this means for `mpf_class' a copy constructor is not
|
||
the same as a default constructor plus assignment.
|
||
|
||
mpf_class x (y); // x created with precision of y
|
||
|
||
mpf_class x; // x created with default precision
|
||
x = y; // value truncated to that precision
|
||
|
||
Applications using templated code may need to be careful about the
|
||
assumptions the code makes in this area, when working with
|
||
`mpf_class' values of various different or non-default precisions.
|
||
For instance implementations of the standard `complex' template
|
||
have been seen in both styles above, though of course `complex' is
|
||
normally only actually specified for use with the builtin float
|
||
types.
|
||
|
||
-- Function: mpf_class abs (mpf_class OP)
|
||
-- Function: mpf_class ceil (mpf_class OP)
|
||
-- Function: int cmp (mpf_class OP1, type OP2)
|
||
-- Function: int cmp (type OP1, mpf_class OP2)
|
||
-- Function: bool mpf_class::fits_sint_p (void)
|
||
-- Function: bool mpf_class::fits_slong_p (void)
|
||
-- Function: bool mpf_class::fits_sshort_p (void)
|
||
-- Function: bool mpf_class::fits_uint_p (void)
|
||
-- Function: bool mpf_class::fits_ulong_p (void)
|
||
-- Function: bool mpf_class::fits_ushort_p (void)
|
||
-- Function: mpf_class floor (mpf_class OP)
|
||
-- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2)
|
||
-- Function: double mpf_class::get_d (void)
|
||
-- Function: long mpf_class::get_si (void)
|
||
-- Function: string mpf_class::get_str (mp_exp_t& EXP, int BASE = 10,
|
||
size_t DIGITS = 0)
|
||
-- Function: unsigned long mpf_class::get_ui (void)
|
||
-- Function: int mpf_class::set_str (const char *STR, int BASE)
|
||
-- Function: int mpf_class::set_str (const string& STR, int BASE)
|
||
-- Function: int sgn (mpf_class OP)
|
||
-- Function: mpf_class sqrt (mpf_class OP)
|
||
-- Function: mpf_class trunc (mpf_class OP)
|
||
These functions provide a C++ class interface to the corresponding
|
||
MPIR C routines.
|
||
|
||
`cmp' can be used with any of the classes or the standard C++
|
||
types, except `long long' and `long double'.
|
||
|
||
The accuracy provided by `hypot' is not currently guaranteed.
|
||
|
||
-- Function: unsigned long int mpf_class::get_prec ()
|
||
-- Function: void mpf_class::set_prec (unsigned long PREC)
|
||
-- Function: void mpf_class::set_prec_raw (unsigned long PREC)
|
||
Get or set the current precision of an `mpf_class'.
|
||
|
||
The restrictions described for `mpf_set_prec_raw' (*note
|
||
Initializing Floats::) apply to `mpf_class::set_prec_raw'. Note
|
||
in particular that the `mpf_class' must be restored to it's
|
||
allocated precision before being destroyed. This must be done by
|
||
application code, there's no automatic mechanism for it.
|
||
|
||
|
||
File: mpir.info, Node: C++ Interface Random Numbers, Next: C++ Interface Limitations, Prev: C++ Interface Floats, Up: C++ Class Interface
|
||
|
||
12.5 C++ Interface Random Numbers
|
||
=================================
|
||
|
||
-- Class: gmp_randclass
|
||
The C++ class interface to the MPIR random number functions uses
|
||
`gmp_randclass' to hold an algorithm selection and current state,
|
||
as per `gmp_randstate_t'.
|
||
|
||
-- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
|
||
(gmp_randstate_t, ...), ...)
|
||
Construct a `gmp_randclass', using a call to the given RANDINIT
|
||
function (*note Random State Initialization::). The arguments
|
||
expected are the same as RANDINIT, but with `mpz_class' instead of
|
||
`mpz_t'. For example,
|
||
|
||
gmp_randclass r1 (gmp_randinit_default);
|
||
gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
|
||
gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
|
||
gmp_randclass r4 (gmp_randinit_mt);
|
||
|
||
`gmp_randinit_lc_2exp_size' will fail if the size requested is too
|
||
big, an `std::length_error' exception is thrown in that case.
|
||
|
||
-- Function: gmp_randclass::gmp_randclass (gmp_randalg_t ALG, ...)
|
||
Construct a `gmp_randclass' using the same parameters as
|
||
`gmp_randinit' (*note Random State Initialization::). This
|
||
function is obsolete and the above RANDINIT style should be
|
||
preferred.
|
||
|
||
-- Function: void gmp_randclass::seed (unsigned long int S)
|
||
-- Function: void gmp_randclass::seed (mpz_class S)
|
||
Seed a random number generator. See *note Random Number
|
||
Functions::, for how to choose a good seed.
|
||
|
||
-- Function: mpz_class gmp_randclass::get_z_bits (unsigned long BITS)
|
||
-- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
|
||
Generate a random integer with a specified number of bits.
|
||
|
||
-- Function: mpz_class gmp_randclass::get_z_range (mpz_class N)
|
||
Generate a random integer in the range 0 to N-1 inclusive.
|
||
|
||
-- Function: mpf_class gmp_randclass::get_f ()
|
||
-- Function: mpf_class gmp_randclass::get_f (unsigned long PREC)
|
||
Generate a random float F in the range 0 <= F < 1. F will be to
|
||
PREC bits precision, or if PREC is not given then to the precision
|
||
of the destination. For example,
|
||
|
||
gmp_randclass r;
|
||
...
|
||
mpf_class f (0, 512); // 512 bits precision
|
||
f = r.get_f(); // random number, 512 bits
|
||
|
||
|
||
File: mpir.info, Node: C++ Interface Limitations, Prev: C++ Interface Random Numbers, Up: C++ Class Interface
|
||
|
||
12.6 C++ Interface Limitations
|
||
==============================
|
||
|
||
`mpq_class' and Templated Reading
|
||
A generic piece of template code probably won't know that
|
||
`mpq_class' requires a `canonicalize' call if inputs read with
|
||
`operator>>' might be non-canonical. This can lead to incorrect
|
||
results.
|
||
|
||
`operator>>' behaves as it does for reasons of efficiency. A
|
||
canonicalize can be quite time consuming on large operands, and is
|
||
best avoided if it's not necessary.
|
||
|
||
But this potential difficulty reduces the usefulness of
|
||
`mpq_class'. Perhaps a mechanism to tell `operator>>' what to do
|
||
will be adopted in the future, maybe a preprocessor define, a
|
||
global flag, or an `ios' flag pressed into service. Or maybe, at
|
||
the risk of inconsistency, the `mpq_class' `operator>>' could
|
||
canonicalize and leave `mpq_t' `operator>>' not doing so, for use
|
||
on those occasions when that's acceptable. Send feedback or
|
||
alternate ideas to `http://groups.google.com/group/mpir-devel'.
|
||
|
||
Subclassing
|
||
Subclassing the MPIR C++ classes works, but is not currently
|
||
recommended.
|
||
|
||
Expressions involving subclasses resolve correctly (or seem to),
|
||
but in normal C++ fashion the subclass doesn't inherit
|
||
constructors and assignments. There's many of those in the MPIR
|
||
classes, and a good way to reestablish them in a subclass is not
|
||
yet provided.
|
||
|
||
Templated Expressions
|
||
A subtle difficulty exists when using expressions together with
|
||
application-defined template functions. Consider the following,
|
||
with `T' intended to be some numeric type,
|
||
|
||
template <class T>
|
||
T fun (const T &, const T &);
|
||
|
||
When used with, say, plain `mpz_class' variables, it works fine:
|
||
`T' is resolved as `mpz_class'.
|
||
|
||
mpz_class f(1), g(2);
|
||
fun (f, g); // Good
|
||
|
||
But when one of the arguments is an expression, it doesn't work.
|
||
|
||
mpz_class f(1), g(2), h(3);
|
||
fun (f, g+h); // Bad
|
||
|
||
This is because `g+h' ends up being a certain expression template
|
||
type internal to `mpirxx.h', which the C++ template resolution
|
||
rules are unable to automatically convert to `mpz_class'. The
|
||
workaround is simply to add an explicit cast.
|
||
|
||
mpz_class f(1), g(2), h(3);
|
||
fun (f, mpz_class(g+h)); // Good
|
||
|
||
Similarly, within `fun' it may be necessary to cast an expression
|
||
to type `T' when calling a templated `fun2'.
|
||
|
||
template <class T>
|
||
void fun (T f, T g)
|
||
{
|
||
fun2 (f, f+g); // Bad
|
||
}
|
||
|
||
template <class T>
|
||
void fun (T f, T g)
|
||
{
|
||
fun2 (f, T(f+g)); // Good
|
||
}
|
||
|
||
|
||
File: mpir.info, Node: BSD Compatible Functions, Next: Custom Allocation, Prev: C++ Class Interface, Up: Top
|
||
|
||
13 Berkeley MP Compatible Functions
|
||
***********************************
|
||
|
||
These functions are intended to be fully compatible with the Berkeley MP
|
||
library which is available on many BSD derived U*ix systems. The
|
||
`--enable-mpbsd' option must be used when building MPIR to make these
|
||
available (*note Installing MPIR::).
|
||
|
||
The original Berkeley MP library has a usage restriction: you cannot
|
||
use the same variable as both source and destination in a single
|
||
function call. The compatible functions in MPIR do not share this
|
||
restriction--inputs and outputs may overlap.
|
||
|
||
It is not recommended that new programs are written using these
|
||
functions. Apart from the incomplete set of functions, the interface
|
||
for initializing `MINT' objects is more error prone, and the `pow'
|
||
function collides with `pow' in `libm.a'.
|
||
|
||
Include the header `mp.h' to get the definition of the necessary
|
||
types and functions. If you are on a BSD derived system, make sure to
|
||
include GNU `mp.h' if you are going to link the GNU `libmp.a' to your
|
||
program. This means that you probably need to give the `-I<dir>'
|
||
option to the compiler, where `<dir>' is the directory where you have
|
||
GNU `mp.h'.
|
||
|
||
-- Function: MINT * itom (signed short int INITIAL_VALUE)
|
||
Allocate an integer consisting of a `MINT' object and dynamic limb
|
||
space. Initialize the integer to INITIAL_VALUE. Return a pointer
|
||
to the `MINT' object.
|
||
|
||
-- Function: MINT * xtom (char *INITIAL_VALUE)
|
||
Allocate an integer consisting of a `MINT' object and dynamic limb
|
||
space. Initialize the integer from INITIAL_VALUE, a hexadecimal,
|
||
null-terminated C string. Return a pointer to the `MINT' object.
|
||
|
||
-- Function: void move (MINT *SRC, MINT *DEST)
|
||
Set DEST to SRC by copying. Both variables must be previously
|
||
initialized.
|
||
|
||
-- Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
|
||
Add SRC_1 and SRC_2 and put the sum in DESTINATION.
|
||
|
||
-- Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
|
||
Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.
|
||
|
||
-- Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
|
||
Multiply SRC_1 and SRC_2 and put the product in DESTINATION.
|
||
|
||
-- Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT,
|
||
MINT *REMAINDER)
|
||
-- Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT
|
||
*QUOTIENT, signed short int *REMAINDER)
|
||
Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod
|
||
DIVISOR. The quotient is rounded towards zero; the remainder has
|
||
the same sign as the dividend unless it is zero.
|
||
|
||
Some implementations of these functions work differently--or not
|
||
at all--for negative arguments.
|
||
|
||
-- Function: void msqrt (MINT *OP, MINT *ROOT, MINT *REMAINDER)
|
||
Set ROOT to the truncated integer part of the square root of OP,
|
||
like `mpz_sqrt'. Set REMAINDER to OP-ROOT*ROOT, i.e. zero if OP
|
||
is a perfect square.
|
||
|
||
If ROOT and REMAINDER are the same variable, the results are
|
||
undefined.
|
||
|
||
-- Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)
|
||
Set DEST to (BASE raised to EXP) modulo MOD.
|
||
|
||
Note that the name `pow' clashes with `pow' from the standard C
|
||
math library (*note Exponentiation and Logarithms: (libc)Exponents
|
||
and Logarithms.). An application will only be able to use one or
|
||
the other.
|
||
|
||
-- Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)
|
||
Set DEST to BASE raised to EXP.
|
||
|
||
-- Function: void gcd (MINT *OP1, MINT *OP2, MINT *RES)
|
||
Set RES to the greatest common divisor of OP1 and OP2.
|
||
|
||
-- Function: int mcmp (MINT *OP1, MINT *OP2)
|
||
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
|
||
if OP1 = OP2, and a negative value if OP1 < OP2.
|
||
|
||
-- Function: void min (MINT *DEST)
|
||
Input a decimal string from `stdin', and put the read integer in
|
||
DEST. SPC and TAB are allowed in the number string, and are
|
||
ignored.
|
||
|
||
-- Function: void mout (MINT *SRC)
|
||
Output SRC to `stdout', as a decimal string. Also output a
|
||
newline.
|
||
|
||
-- Function: char * mtox (MINT *OP)
|
||
Convert OP to a hexadecimal string, and return a pointer to the
|
||
string. The returned string is allocated using the default memory
|
||
allocation function, `malloc' by default. It will be
|
||
`strlen(str)+1' bytes, that being exactly enough for the string
|
||
and null-terminator.
|
||
|
||
-- Function: void mfree (MINT *OP)
|
||
De-allocate, the space used by OP. *This function should only be
|
||
passed a value returned by `itom' or `xtom'.*
|
||
|
||
|
||
File: mpir.info, Node: Custom Allocation, Next: Language Bindings, Prev: BSD Compatible Functions, Up: Top
|
||
|
||
14 Custom Allocation
|
||
********************
|
||
|
||
By default MPIR uses `malloc', `realloc' and `free' for memory
|
||
allocation, and if they fail MPIR prints a message to the standard
|
||
error output and terminates the program.
|
||
|
||
Alternate functions can be specified, to allocate memory in a
|
||
different way or to have a different error action on running out of
|
||
memory.
|
||
|
||
This feature is available in the Berkeley compatibility library
|
||
(*note BSD Compatible Functions::) as well as the main MPIR library.
|
||
|
||
-- Function: void mp_set_memory_functions (
|
||
void *(*ALLOC_FUNC_PTR) (size_t),
|
||
void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
|
||
void (*FREE_FUNC_PTR) (void *, size_t))
|
||
Replace the current allocation functions from the arguments. If
|
||
an argument is `NULL', the corresponding default function is used.
|
||
|
||
These functions will be used for all memory allocation done by
|
||
MPIR, apart from temporary space from `alloca' if that function is
|
||
available and MPIR is configured to use it (*note Build Options::).
|
||
|
||
*Be sure to call `mp_set_memory_functions' only when there are no
|
||
active MPIR objects allocated using the previous memory functions!
|
||
Usually that means calling it before any other MPIR function.*
|
||
|
||
The functions supplied should fit the following declarations:
|
||
|
||
-- Function: void * allocate_function (size_t ALLOC_SIZE)
|
||
Return a pointer to newly allocated space with at least ALLOC_SIZE
|
||
bytes.
|
||
|
||
-- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
|
||
size_t NEW_SIZE)
|
||
Resize a previously allocated block PTR of OLD_SIZE bytes to be
|
||
NEW_SIZE bytes.
|
||
|
||
The block may be moved if necessary or if desired, and in that
|
||
case the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to
|
||
the new location. The return value is a pointer to the resized
|
||
block, that being the new location if moved or just PTR if not.
|
||
|
||
PTR is never `NULL', it's always a previously allocated block.
|
||
NEW_SIZE may be bigger or smaller than OLD_SIZE.
|
||
|
||
-- Function: void free_function (void *PTR, size_t SIZE)
|
||
De-allocate the space pointed to by PTR.
|
||
|
||
PTR is never `NULL', it's always a previously allocated block of
|
||
SIZE bytes.
|
||
|
||
A "byte" here means the unit used by the `sizeof' operator.
|
||
|
||
The OLD_SIZE parameters to REALLOCATE_FUNCTION and FREE_FUNCTION are
|
||
passed for convenience, but of course can be ignored if not needed.
|
||
The default functions using `malloc' and friends for instance don't use
|
||
them.
|
||
|
||
No error return is allowed from any of these functions, if they
|
||
return then they must have performed the specified operation. In
|
||
particular note that ALLOCATE_FUNCTION or REALLOCATE_FUNCTION mustn't
|
||
return `NULL'.
|
||
|
||
Getting a different fatal error action is a good use for custom
|
||
allocation functions, for example giving a graphical dialog rather than
|
||
the default print to `stderr'. How much is possible when genuinely out
|
||
of memory is another question though.
|
||
|
||
There's currently no defined way for the allocation functions to
|
||
recover from an error such as out of memory, they must terminate
|
||
program execution. A `longjmp' or throwing a C++ exception will have
|
||
undefined results. This may change in the future.
|
||
|
||
MPIR may use allocated blocks to hold pointers to other allocated
|
||
blocks. This will limit the assumptions a conservative garbage
|
||
collection scheme can make.
|
||
|
||
Since the default MPIR allocation uses `malloc' and friends, those
|
||
functions will be linked in even if the first thing a program does is an
|
||
`mp_set_memory_functions'. It's necessary to change the MPIR sources if
|
||
this is a problem.
|
||
|
||
|
||
-- Function: void mp_get_memory_functions (
|
||
void *(**ALLOC_FUNC_PTR) (size_t),
|
||
void *(**REALLOC_FUNC_PTR) (void *, size_t, size_t),
|
||
void (**FREE_FUNC_PTR) (void *, size_t))
|
||
Get the current allocation functions, storing function pointers to
|
||
the locations given by the arguments. If an argument is `NULL',
|
||
that function pointer is not stored.
|
||
|
||
For example, to get just the current free function,
|
||
|
||
void (*freefunc) (void *, size_t);
|
||
|
||
mp_get_memory_functions (NULL, NULL, &freefunc);
|
||
|
||
|
||
File: mpir.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
|
||
|
||
15 Language Bindings
|
||
********************
|
||
|
||
The following packages and projects offer access to MPIR from languages
|
||
other than C, though perhaps with varying levels of functionality and
|
||
efficiency.
|
||
|
||
|
||
C++
|
||
* MPIR C++ class interface, *note C++ Class Interface::
|
||
Straightforward interface, expression templates to eliminate
|
||
temporaries.
|
||
|
||
* ALP `http://www-sop.inria.fr/saga/logiciels/ALP/'
|
||
Linear algebra and polynomials using templates.
|
||
|
||
* Arithmos `http://www.win.ua.ac.be/~cant/arithmos/'
|
||
Rationals with infinities and square roots.
|
||
|
||
* CLN `http://www.ginac.de/CLN/'
|
||
High level classes for arithmetic.
|
||
|
||
* LiDIA `http://www.informatik.tu-darmstadt.de/TI/LiDIA/'
|
||
A C++ library for computational number theory.
|
||
|
||
* Linbox `http://www.linalg.org/'
|
||
Sparse vectors and matrices.
|
||
|
||
* NTL `http://www.shoup.net/ntl/'
|
||
A C++ number theory library.
|
||
|
||
D
|
||
* gmp-d `http://home.comcast.net/~benhinkle/gmp-d'
|
||
|
||
Fortran
|
||
* Omni F77 `http://phase.hpcc.jp/Omni/home.html'
|
||
Arbitrary precision floats.
|
||
|
||
Haskell
|
||
* Glasgow Haskell Compiler `http://www.haskell.org/ghc/'
|
||
|
||
Java
|
||
* Kaffe `http://www.kaffe.org/'
|
||
|
||
* Kissme `http://kissme.sourceforge.net/'
|
||
|
||
Lisp
|
||
* GNU Common Lisp `http://www.gnu.org/software/gcl/gcl.html'
|
||
|
||
* Librep `http://librep.sourceforge.net/'
|
||
|
||
* XEmacs (21.5.18 beta and up) `http://www.xemacs.org'
|
||
Optional big integers, rationals and floats using MPIR.
|
||
|
||
M4
|
||
* GNU m4 betas `http://www.seindal.dk/rene/gnu/'
|
||
Optionally provides an arbitrary precision `mpeval'.
|
||
|
||
ML
|
||
* MLton compiler `http://mlton.org/'
|
||
|
||
Objective Caml
|
||
* MLGMP `http://www.di.ens.fr/~monniaux/programmes.html.en'
|
||
|
||
* Numerix `http://pauillac.inria.fr/~quercia/'
|
||
Optionally using GMP.
|
||
|
||
Oz
|
||
* Mozart `http://www.mozart-oz.org/'
|
||
|
||
Pascal
|
||
* GNU Pascal Compiler `http://www.gnu-pascal.de/'
|
||
GMP unit.
|
||
|
||
* Numerix `http://pauillac.inria.fr/~quercia/'
|
||
For Free Pascal, optionally using GMP.
|
||
|
||
Perl
|
||
* GMP module, see `demos/perl' in the MPIR sources (*note
|
||
Demonstration Programs::).
|
||
|
||
* Math::GMP `http://www.cpan.org/'
|
||
Compatible with Math::BigInt, but not as many functions as
|
||
the GMP module above.
|
||
|
||
* Math::BigInt::GMP `http://www.cpan.org/'
|
||
Plug Math::GMP into normal Math::BigInt operations.
|
||
|
||
Pike
|
||
* mpz module in the standard distribution,
|
||
`http://pike.ida.liu.se/'
|
||
|
||
Prolog
|
||
* SWI Prolog `http://www.swi-prolog.org/'
|
||
Arbitrary precision floats.
|
||
|
||
Python
|
||
* mpz module in the standard distribution,
|
||
`http://www.python.org/'
|
||
|
||
* GMPY `http://gmpy.sourceforge.net/'
|
||
|
||
Scheme
|
||
* GNU Guile (upcoming 1.8)
|
||
`http://www.gnu.org/software/guile/guile.html'
|
||
|
||
* RScheme `http://www.rscheme.org/'
|
||
|
||
* STklos `http://www.stklos.org/'
|
||
|
||
Smalltalk
|
||
* GNU Smalltalk
|
||
`http://www.smalltalk.org/versions/GNUSmalltalk.html'
|
||
|
||
Other
|
||
* Axiom `http://savannah.nongnu.org/projects/axiom'
|
||
Computer algebra using GCL.
|
||
|
||
* DrGenius `http://drgenius.seul.org/'
|
||
Geometry system and mathematical programming language.
|
||
|
||
* GiNaC `http://www.ginac.de/'
|
||
C++ computer algebra using CLN.
|
||
|
||
* GOO `http://www.googoogaga.org/'
|
||
Dynamic object oriented language.
|
||
|
||
* Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html'
|
||
Macsyma computer algebra using GCL.
|
||
|
||
* Q `http://q-lang.sourceforge.net/'
|
||
Equational programming system.
|
||
|
||
* Regina `http://regina.sourceforge.net/'
|
||
Topological calculator.
|
||
|
||
* Yacas `http://www.xs4all.nl/~apinkus/yacas.html'
|
||
Yet another computer algebra system.
|
||
|
||
|
||
|
||
File: mpir.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top
|
||
|
||
16 Algorithms
|
||
*************
|
||
|
||
This chapter is an introduction to some of the algorithms used for
|
||
various MPIR operations. The code is likely to be hard to understand
|
||
without knowing something about the algorithms.
|
||
|
||
Some MPIR internals are mentioned, but applications that expect to be
|
||
compatible with future MPIR releases should take care to use only the
|
||
documented functions.
|
||
|
||
* Menu:
|
||
|
||
* Multiplication Algorithms::
|
||
* Division Algorithms::
|
||
* Greatest Common Divisor Algorithms::
|
||
* Powering Algorithms::
|
||
* Root Extraction Algorithms::
|
||
* Radix Conversion Algorithms::
|
||
* Other Algorithms::
|
||
* Assembler Coding::
|
||
|
||
|
||
File: mpir.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
|
||
|
||
16.1 Multiplication
|
||
===================
|
||
|
||
NxN limb multiplications and squares are done using one of four
|
||
algorithms, as the size N increases.
|
||
|
||
Algorithm Threshold
|
||
Basecase (none)
|
||
Karatsuba `MUL_KARATSUBA_THRESHOLD'
|
||
Toom-3 `MUL_TOOM3_THRESHOLD'
|
||
FFT `MUL_FFT_THRESHOLD'
|
||
|
||
Similarly for squaring, with the `SQR' thresholds.
|
||
|
||
NxM multiplications of operands with different sizes above
|
||
`MUL_KARATSUBA_THRESHOLD' are currently done by splitting into MxM
|
||
pieces. The Karatsuba and Toom-3 routines then operate only on equal
|
||
size operands. This is not very efficient, and is slated for
|
||
improvement in the future.
|
||
|
||
* Menu:
|
||
|
||
* Basecase Multiplication::
|
||
* Karatsuba Multiplication::
|
||
* Toom 3-Way Multiplication::
|
||
* FFT Multiplication::
|
||
* Other Multiplication::
|
||
|
||
|
||
File: mpir.info, Node: Basecase Multiplication, Next: Karatsuba Multiplication, Prev: Multiplication Algorithms, Up: Multiplication Algorithms
|
||
|
||
16.1.1 Basecase Multiplication
|
||
------------------------------
|
||
|
||
Basecase NxM multiplication is a straightforward rectangular set of
|
||
cross-products, the same as long multiplication done by hand and for
|
||
that reason sometimes known as the schoolbook or grammar school method.
|
||
This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M
|
||
(*note References::), and the `mpn/generic/mul_basecase.c' code.
|
||
|
||
Assembler implementations of `mpn_mul_basecase' are essentially the
|
||
same as the generic C code, but have all the usual assembler tricks and
|
||
obscurities introduced for speed.
|
||
|
||
A square can be done in roughly half the time of a multiply, by
|
||
using the fact that the cross products above and below the diagonal are
|
||
the same. A triangle of products below the diagonal is formed, doubled
|
||
(left shift by one bit), and then the products on the diagonal added.
|
||
This can be seen in `mpn/generic/sqr_basecase.c'. Again the assembler
|
||
implementations take essentially the same approach.
|
||
|
||
u0 u1 u2 u3 u4
|
||
+---+---+---+---+---+
|
||
u0 | d | | | | |
|
||
+---+---+---+---+---+
|
||
u1 | | d | | | |
|
||
+---+---+---+---+---+
|
||
u2 | | | d | | |
|
||
+---+---+---+---+---+
|
||
u3 | | | | d | |
|
||
+---+---+---+---+---+
|
||
u4 | | | | | d |
|
||
+---+---+---+---+---+
|
||
|
||
In practice squaring isn't a full 2x faster than multiplying, it's
|
||
usually around 1.5x. Less than 1.5x probably indicates
|
||
`mpn_sqr_basecase' wants improving on that CPU.
|
||
|
||
On some CPUs `mpn_mul_basecase' can be faster than the generic C
|
||
`mpn_sqr_basecase' on some small sizes. `SQR_BASECASE_THRESHOLD' is
|
||
the size at which to use `mpn_sqr_basecase', this will be zero if that
|
||
routine should be used always.
|
||
|
||
|
||
File: mpir.info, Node: Karatsuba Multiplication, Next: Toom 3-Way Multiplication, Prev: Basecase Multiplication, Up: Multiplication Algorithms
|
||
|
||
16.1.2 Karatsuba Multiplication
|
||
-------------------------------
|
||
|
||
The Karatsuba multiplication algorithm is described in Knuth section
|
||
4.3.3 part A, and various other textbooks. A brief description is
|
||
given here.
|
||
|
||
The inputs x and y are treated as each split into two parts of equal
|
||
length (or the most significant part one limb shorter if N is odd).
|
||
|
||
high low
|
||
+----------+----------+
|
||
| x1 | x0 |
|
||
+----------+----------+
|
||
|
||
+----------+----------+
|
||
| y1 | y0 |
|
||
+----------+----------+
|
||
|
||
Let b be the power of 2 where the split occurs, ie. if x0 is k limbs
|
||
(y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and
|
||
y=y1*b+y0, and the following holds,
|
||
|
||
x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
|
||
|
||
This formula means doing only three multiplies of (N/2)x(N/2) limbs,
|
||
whereas a basecase multiply of NxN limbs is equivalent to four
|
||
multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
|
||
positions where the three products must be added.
|
||
|
||
high low
|
||
+--------+--------+ +--------+--------+
|
||
| x1*y1 | | x0*y0 |
|
||
+--------+--------+ +--------+--------+
|
||
+--------+--------+
|
||
add | x1*y1 |
|
||
+--------+--------+
|
||
+--------+--------+
|
||
add | x0*y0 |
|
||
+--------+--------+
|
||
+--------+--------+
|
||
sub | (x1-x0)*(y1-y0) |
|
||
+--------+--------+
|
||
|
||
The term (x1-x0)*(y1-y0) is best calculated as an absolute value,
|
||
and the sign used to choose to add or subtract. Notice the sum
|
||
high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
|
||
additions, rather than 6*k, but in MPIR extra function call overheads
|
||
outweigh the saving.
|
||
|
||
Squaring is similar to multiplying, but with x=y the formula reduces
|
||
to an equivalent with three squares,
|
||
|
||
x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
|
||
|
||
The final result is accumulated from those three squares the same
|
||
way as for the three multiplies above. The middle term (x1-x0)^2 is now
|
||
always positive.
|
||
|
||
A similar formula for both multiplying and squaring can be
|
||
constructed with a middle term (x1+x0)*(y1+y0). But those sums can
|
||
exceed k limbs, leading to more carry handling and additions than the
|
||
form above.
|
||
|
||
Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
|
||
the exponent being log(3)/log(2), representing 3 multiplies each 1/2
|
||
the size of the inputs. This is a big improvement over the basecase
|
||
multiply at O(N^2) and the advantage soon overcomes the extra additions
|
||
Karatsuba performs. `MUL_KARATSUBA_THRESHOLD' can be as little as 10
|
||
limbs. The `SQR' threshold is usually about twice the `MUL'.
|
||
|
||
The basecase algorithm will take a time of the form M(N) = a*N^2 +
|
||
b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, which
|
||
expands to K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e. The factor 3/4
|
||
for a means per-crossproduct speedups in the basecase code will
|
||
increase the threshold since they benefit M(N) more than K(N). And
|
||
conversely the 3/2 for b means linear style speedups of b will increase
|
||
the threshold since they benefit K(N) more than M(N). The latter can
|
||
be seen for instance when adding an optimized `mpn_sqr_diagonal' to
|
||
`mpn_sqr_basecase'. Of course all speedups reduce total time, and in
|
||
that sense the algorithm thresholds are merely of academic interest.
|
||
|
||
|
||
File: mpir.info, Node: Toom 3-Way Multiplication, Next: FFT Multiplication, Prev: Karatsuba Multiplication, Up: Multiplication Algorithms
|
||
|
||
16.1.3 Toom 3-Way Multiplication
|
||
--------------------------------
|
||
|
||
The Karatsuba formula is the simplest case of a general approach to
|
||
splitting inputs that leads to both Toom and FFT algorithms. A
|
||
description of Toom can be found in Knuth section 4.3.3, with an
|
||
example 3-way calculation after Theorem A. The 3-way form used in MPIR
|
||
is described here.
|
||
|
||
The operands are each considered split into 3 pieces of equal length
|
||
(or the most significant part 1 or 2 limbs shorter than the other two).
|
||
|
||
high low
|
||
+----------+----------+----------+
|
||
| x2 | x1 | x0 |
|
||
+----------+----------+----------+
|
||
|
||
+----------+----------+----------+
|
||
| y2 | y1 | y0 |
|
||
+----------+----------+----------+
|
||
|
||
These parts are treated as the coefficients of two polynomials
|
||
|
||
X(t) = x2*t^2 + x1*t + x0
|
||
Y(t) = y2*t^2 + y1*t + y0
|
||
|
||
Let b equal the power of 2 which is the size of the x0, x1, y0 and
|
||
y1 pieces, ie. if they're k limbs each then b=2^(k*mp_bits_per_limb).
|
||
With this x=X(b) and y=Y(b).
|
||
|
||
Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
|
||
|
||
W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
|
||
|
||
The w[i] are going to be determined, and when they are they'll give
|
||
the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The
|
||
coefficients will be roughly b^2 each, and the final W(b) will be an
|
||
addition like,
|
||
|
||
high low
|
||
+-------+-------+
|
||
| w4 |
|
||
+-------+-------+
|
||
+--------+-------+
|
||
| w3 |
|
||
+--------+-------+
|
||
+--------+-------+
|
||
| w2 |
|
||
+--------+-------+
|
||
+--------+-------+
|
||
| w1 |
|
||
+--------+-------+
|
||
+-------+-------+
|
||
| w0 |
|
||
+-------+-------+
|
||
|
||
The w[i] coefficients could be formed by a simple set of cross
|
||
products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
|
||
this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
|
||
equivalent merely to a basecase multiply. Instead the following
|
||
approach is used.
|
||
|
||
X(t) and Y(t) are evaluated and multiplied at 5 points, giving
|
||
values of W(t) at those points. In MPIR the following points are used,
|
||
|
||
Point Value
|
||
t=0 x0 * y0, which gives w0 immediately
|
||
t=1 (x2+x1+x0) * (y2+y1+y0)
|
||
t=-1 (x2-x1+x0) * (y2-y1+y0)
|
||
t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0)
|
||
t=inf x2 * y2, which gives w4 immediately
|
||
|
||
At t=-1 the values can be negative and that's handled using the
|
||
absolute values and tracking the sign separately. At t=inf the value
|
||
is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but
|
||
it's much easier to think of as simply x2*y2 giving w4 immediately
|
||
(much like x0*y0 at t=0 gives w0 immediately).
|
||
|
||
Each of the points substituted into W(t)=w4*t^4+...+w0 gives a
|
||
linear combination of the w[i] coefficients, and the value of those
|
||
combinations has just been calculated.
|
||
|
||
W(0) = w0
|
||
W(1) = w4 + w3 + w2 + w1 + w0
|
||
W(-1) = w4 - w3 + w2 - w1 + w0
|
||
W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
|
||
W(inf) = w4
|
||
|
||
This is a set of five equations in five unknowns, and some
|
||
elementary linear algebra quickly isolates each w[i]. This involves
|
||
adding or subtracting one W(t) value from another, and a couple of
|
||
divisions by powers of 2 and one division by 3, the latter using the
|
||
special `mpn_divexact_by3' (*note Exact Division::).
|
||
|
||
The conversion of W(t) values to the coefficients is interpolation.
|
||
A polynomial of degree 4 like W(t) is uniquely determined by values
|
||
known at 5 different points. The points are arbitrary and can be
|
||
chosen to make the linear equations come out with a convenient set of
|
||
steps for quickly isolating the w[i].
|
||
|
||
Squaring follows the same procedure as multiplication, but there's
|
||
only one X(t) and it's evaluated at the 5 points, and those values
|
||
squared to give values of W(t). The interpolation is then identical,
|
||
and in fact the same `toom3_interpolate' subroutine is used for both
|
||
squaring and multiplying.
|
||
|
||
Toom-3 is asymptotically O(N^1.465), the exponent being
|
||
log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
|
||
size each. This is an improvement over Karatsuba at O(N^1.585), though
|
||
Toom does more work in the evaluation and interpolation and so it only
|
||
realizes its advantage above a certain size.
|
||
|
||
Near the crossover between Toom-3 and Karatsuba there's generally a
|
||
range of sizes where the difference between the two is small.
|
||
`MUL_TOOM3_THRESHOLD' is a somewhat arbitrary point in that range and
|
||
successive runs of the tune program can give different values due to
|
||
small variations in measuring. A graph of time versus size for the two
|
||
shows the effect, see `tune/README'.
|
||
|
||
At the fairly small sizes where the Toom-3 thresholds occur it's
|
||
worth remembering that the asymptotic behaviour for Karatsuba and
|
||
Toom-3 can't be expected to make accurate predictions, due of course to
|
||
the big influence of all sorts of overheads, and the fact that only a
|
||
few recursions of each are being performed. Even at large sizes
|
||
there's a good chance machine dependent effects like cache architecture
|
||
will mean actual performance deviates from what might be predicted.
|
||
|
||
The formula given for the Karatsuba algorithm (*note Karatsuba
|
||
Multiplication::) has an equivalent for Toom-3 involving only five
|
||
multiplies, but this would be complicated and unenlightening.
|
||
|
||
An alternate view of Toom-3 can be found in Zuras (*note
|
||
References::), using a vector to represent the x and y splits and a
|
||
matrix multiplication for the evaluation and interpolation stages. The
|
||
matrix inverses are not meant to be actually used, and they have
|
||
elements with values much greater than in fact arise in the
|
||
interpolation steps. The diagram shown for the 3-way is attractive,
|
||
but again doesn't have to be implemented that way and for example with
|
||
a bit of rearrangement just one division by 6 can be done.
|
||
|
||
|
||
File: mpir.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
|
||
|
||
16.1.4 FFT Multiplication
|
||
-------------------------
|
||
|
||
At large to very large sizes a Fermat style FFT multiplication is used,
|
||
following Scho"nhage and Strassen (*note References::). Descriptions
|
||
of FFTs in various forms can be found in many textbooks, for instance
|
||
Knuth section 4.3.3 part C or Lipson chapter IX. A brief description
|
||
of the form used in MPIR is given here.
|
||
|
||
The multiplication done is x*y mod 2^N+1, for a given N. A full
|
||
product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x
|
||
and y with high zero limbs. The modular product is the native form for
|
||
the algorithm, so padding to get a full product is unavoidable.
|
||
|
||
The algorithm follows a split, evaluate, pointwise multiply,
|
||
interpolate and combine similar to that described above for Karatsuba
|
||
and Toom-3. A k parameter controls the split, with an FFT-k splitting
|
||
into 2^k pieces of M=N/2^k bits each. N must be a multiple of
|
||
(2^k)*mp_bits_per_limb so the split falls on limb boundaries, avoiding
|
||
bit shifts in the split and combine stages.
|
||
|
||
The evaluations, pointwise multiplications, and interpolation, are
|
||
all done modulo 2^N'+1 where N' is 2M+k+3 rounded up to a multiple of
|
||
2^k and of `mp_bits_per_limb'. The results of interpolation will be
|
||
the following negacyclic convolution of the input pieces, and the
|
||
choice of N' ensures these sums aren't truncated.
|
||
|
||
---
|
||
\ b
|
||
w[n] = / (-1) * x[i] * y[j]
|
||
---
|
||
i+j==b*2^k+n
|
||
b=0,1
|
||
|
||
The points used for the evaluation are g^i for i=0 to 2^k-1 where
|
||
g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces
|
||
necessary cancellations at the interpolation stage, and it's also a
|
||
power of 2 so the fast fourier transforms used for the evaluation and
|
||
interpolation do only shifts, adds and negations.
|
||
|
||
The pointwise multiplications are done modulo 2^N'+1 and either
|
||
recurse into a further FFT or use a plain multiplication (Toom-3,
|
||
Karatsuba or basecase), whichever is optimal at the size N'. The
|
||
interpolation is an inverse fast fourier transform. The resulting set
|
||
of sums of x[i]*y[j] are added at appropriate offsets to give the final
|
||
result.
|
||
|
||
Squaring is the same, but x is the only input so it's one transform
|
||
at the evaluate stage and the pointwise multiplies are squares. The
|
||
interpolation is the same.
|
||
|
||
For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm,
|
||
the exponent representing 2^k recursed modular multiplies each
|
||
1/2^(k-1) the size of the original. Each successive k is an asymptotic
|
||
improvement, but overheads mean each is only faster at bigger and
|
||
bigger sizes. In the code, `MUL_FFT_TABLE' and `SQR_FFT_TABLE' are the
|
||
thresholds where each k is used. Each new k effectively swaps some
|
||
multiplying for some shifts, adds and overheads.
|
||
|
||
A mod 2^N+1 product can be formed with a normal NxN->2N bit multiply
|
||
plus a subtraction, so an FFT and Toom-3 etc can be compared directly.
|
||
A k=4 FFT at O(N^1.333) can be expected to be the first faster than
|
||
Toom-3 at O(N^1.465). In practice this is what's found, with
|
||
`MUL_FFT_MODF_THRESHOLD' and `SQR_FFT_MODF_THRESHOLD' being between 300
|
||
and 1000 limbs, depending on the CPU. So far it's been found that only
|
||
very large FFTs recurse into pointwise multiplies above these sizes.
|
||
|
||
When an FFT is to give a full product, the change of N to 2N doesn't
|
||
alter the theoretical complexity for a given k, but for the purposes of
|
||
considering where an FFT might be first used it can be assumed that the
|
||
FFT is recursing into a normal multiply and that on that basis it's
|
||
doing 2^k recursed multiplies each 1/2^(k-2) the size of the inputs,
|
||
making it O(N^(k/(k-2))). This would mean k=7 at O(N^1.4) would be the
|
||
first FFT faster than Toom-3. In practice `MUL_FFT_THRESHOLD' and
|
||
`SQR_FFT_THRESHOLD' have been found to be in the k=8 range, somewhere
|
||
between 3000 and 10000 limbs.
|
||
|
||
The way N is split into 2^k pieces and then 2M+k+3 is rounded up to
|
||
a multiple of 2^k and `mp_bits_per_limb' means that when
|
||
2^k>=mp_bits_per_limb the effective N is a multiple of 2^(2k-1) bits.
|
||
The +k+3 means some values of N just under such a multiple will be
|
||
rounded to the next. The complexity calculations above assume that a
|
||
favourable size is used, meaning one which isn't padded through
|
||
rounding, and it's also assumed that the extra +k+3 bits are negligible
|
||
at typical FFT sizes.
|
||
|
||
The practical effect of the 2^(2k-1) constraint is to introduce a
|
||
step-effect into measured speeds. For example k=8 will round N up to a
|
||
multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
|
||
groups of sizes for which `mpn_mul_n' runs at the same speed. Or for
|
||
k=9 groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In practice
|
||
it's been found each k is used at quite small multiples of its size
|
||
constraint and so the step effect is quite noticeable in a time versus
|
||
size graph.
|
||
|
||
The threshold determinations currently measure at the mid-points of
|
||
size steps, but this is sub-optimal since at the start of a new step it
|
||
can happen that it's better to go back to the previous k for a while.
|
||
Something more sophisticated for `MUL_FFT_TABLE' and `SQR_FFT_TABLE'
|
||
will be needed.
|
||
|
||
|
||
File: mpir.info, Node: Other Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms
|
||
|
||
16.1.5 Other Multiplication
|
||
---------------------------
|
||
|
||
The 3-way Toom algorithm described above (*note Toom 3-Way
|
||
Multiplication::) generalizes to split into an arbitrary number of
|
||
pieces, as per Knuth section 4.3.3 algorithm C. This is not currently
|
||
used, though it's possible a Toom-4 might fit in between Toom-3 and the
|
||
FFTs. The notes here are merely for interest.
|
||
|
||
In general a split into r+1 pieces is made, and evaluations and
|
||
pointwise multiplications done at 2*r+1 points. A 4-way split does 7
|
||
pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way
|
||
algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise
|
||
multiplications count towards big-O complexity, but the time spent in
|
||
the evaluate and interpolate stages grows with r and has a significant
|
||
practical impact, with the asymptotic advantage of each r realized only
|
||
at bigger and bigger sizes. The overheads grow as O(N*r), whereas in
|
||
an r=2^k FFT they grow only as O(N*log(r)).
|
||
|
||
Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4
|
||
uses -r,...,0,...,r and the latter saves some small multiplies in the
|
||
evaluate stage (or rather trades them for additions), and has a further
|
||
saving of nearly half the interpolate steps. The idea is to separate
|
||
odd and even final coefficients and then perform algorithm C steps C7
|
||
and C8 on them separately. The divisors at step C7 become j^2 and the
|
||
multipliers at C8 become 2*t*j-j^2.
|
||
|
||
Splitting odd and even parts through positive and negative points
|
||
can be thought of as using -1 as a square root of unity. If a 4th root
|
||
of unity was available then a further split and speedup would be
|
||
possible, but no such root exists for plain integers. Going to complex
|
||
integers with i=sqrt(-1) doesn't help, essentially because in cartesian
|
||
form it takes three real multiplies to do a complex multiply. The
|
||
existence of 2^k'th roots of unity in a suitable ring or field lets the
|
||
fast fourier transform keep splitting and get to O(N*log(r)).
|
||
|
||
Floating point FFTs use complex numbers approximating Nth roots of
|
||
unity. Some processors have special support for such FFTs. But these
|
||
are not used in MPIR since it's very difficult to guarantee an exact
|
||
result (to some number of bits). An occasional difference of 1 in the
|
||
last bit might not matter to a typical signal processing algorithm, but
|
||
is of course of vital importance to MPIR.
|
||
|
||
|
||
File: mpir.info, Node: Division Algorithms, Next: Greatest Common Divisor Algorithms, Prev: Multiplication Algorithms, Up: Algorithms
|
||
|
||
16.2 Division Algorithms
|
||
========================
|
||
|
||
* Menu:
|
||
|
||
* Single Limb Division::
|
||
* Basecase Division::
|
||
* Divide and Conquer Division::
|
||
* Exact Division::
|
||
* Exact Remainder::
|
||
* Small Quotient Division::
|
||
|
||
|
||
File: mpir.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms
|
||
|
||
16.2.1 Single Limb Division
|
||
---------------------------
|
||
|
||
Nx1 division is implemented using repeated 2x1 divisions from high to
|
||
low, either with a hardware divide instruction or a multiplication by
|
||
inverse, whichever is best on a given CPU.
|
||
|
||
The multiply by inverse follows section 8 of "Division by Invariant
|
||
Integers using Multiplication" by Granlund and Montgomery (*note
|
||
References::) and is implemented as `udiv_qrnnd_preinv' in
|
||
`gmp-impl.h'. The idea is to have a fixed-point approximation to 1/d
|
||
(see `invert_limb') and then multiply by the high limb (plus one bit)
|
||
of the dividend to get a quotient q. With d normalized (high bit set),
|
||
q is no more than 1 too small. Subtracting q*d from the dividend gives
|
||
a remainder, and reveals whether q or q-1 is correct.
|
||
|
||
The result is a division done with two multiplications and four or
|
||
five arithmetic operations. On CPUs with low latency multipliers this
|
||
can be much faster than a hardware divide, though the cost of
|
||
calculating the inverse at the start may mean it's only better on
|
||
inputs bigger than say 4 or 5 limbs.
|
||
|
||
When a divisor must be normalized, either for the generic C
|
||
`__udiv_qrnnd_c' or the multiply by inverse, the division performed is
|
||
actually a*2^k by d*2^k where a is the dividend and k is the power
|
||
necessary to have the high bit of d*2^k set. The bit shifts for the
|
||
dividend are usually accomplished "on the fly" meaning by extracting
|
||
the appropriate bits at each step. Done this way the quotient limbs
|
||
come out aligned ready to store. When only the remainder is wanted, an
|
||
alternative is to take the dividend limbs unshifted and calculate r = a
|
||
mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can
|
||
help on CPUs with poor bit shifts or few registers.
|
||
|
||
The multiply by inverse can be done two limbs at a time. The
|
||
calculation is basically the same, but the inverse is two limbs and the
|
||
divisor treated as if padded with a low zero limb. This means more
|
||
work, since the inverse will need a 2x2 multiply, but the four 1x1s to
|
||
do that are independent and can therefore be done partly or wholly in
|
||
parallel. Likewise for a 2x1 calculating q*d. The net effect is to
|
||
process two limbs with roughly the same two multiplies worth of latency
|
||
that one limb at a time gives. This extends to 3 or 4 limbs at a time,
|
||
though the extra work to apply the inverse will almost certainly soon
|
||
reach the limits of multiplier throughput.
|
||
|
||
A similar approach in reverse can be taken to process just half a
|
||
limb at a time if the divisor is only a half limb. In this case the
|
||
1x1 multiply for the inverse effectively becomes two (1/2)x1 for each
|
||
limb, which can be a saving on CPUs with a fast half limb multiply, or
|
||
in fact if the only multiply is a half limb, and especially if it's not
|
||
pipelined.
|
||
|
||
|
||
File: mpir.info, Node: Basecase Division, Next: Divide and Conquer Division, Prev: Single Limb Division, Up: Division Algorithms
|
||
|
||
16.2.2 Basecase Division
|
||
------------------------
|
||
|
||
Basecase NxM division is like long division done by hand, but in base
|
||
2^mp_bits_per_limb. See Knuth section 4.3.1 algorithm D, and
|
||
`mpn/generic/sb_divrem_mn.c'.
|
||
|
||
Briefly stated, while the dividend remains larger than the divisor,
|
||
a high quotient limb is formed and the Nx1 product q*d subtracted at
|
||
the top end of the dividend. With a normalized divisor (most
|
||
significant bit set), each quotient limb can be formed with a 2x1
|
||
division and a 1x1 multiplication plus some subtractions. The 2x1
|
||
division is by the high limb of the divisor and is done either with a
|
||
hardware divide or a multiply by inverse (the same as in *note Single
|
||
Limb Division::) whichever is faster. Such a quotient is sometimes one
|
||
too big, requiring an addback of the divisor, but that happens rarely.
|
||
|
||
With Q=N-M being the number of quotient limbs, this is an O(Q*M)
|
||
algorithm and will run at a speed similar to a basecase QxM
|
||
multiplication, differing in fact only in the extra multiply and divide
|
||
for each of the Q quotient limbs.
|
||
|
||
|
||
File: mpir.info, Node: Divide and Conquer Division, Next: Exact Division, Prev: Basecase Division, Up: Division Algorithms
|
||
|
||
16.2.3 Divide and Conquer Division
|
||
----------------------------------
|
||
|
||
For divisors larger than `DIV_DC_THRESHOLD', division is done by
|
||
dividing. Or to be precise by a recursive divide and conquer algorithm
|
||
based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler
|
||
(*note References::).
|
||
|
||
The algorithm consists essentially of recognising that a 2NxN
|
||
division can be done with the basecase division algorithm (*note
|
||
Basecase Division::), but using N/2 limbs as a base, not just a single
|
||
limb. This way the multiplications that arise are (N/2)x(N/2) and can
|
||
take advantage of Karatsuba and higher multiplication algorithms (*note
|
||
Multiplication Algorithms::). The two "digits" of the quotient are
|
||
formed by recursive Nx(N/2) divisions.
|
||
|
||
If the (N/2)x(N/2) multiplies are done with a basecase multiplication
|
||
then the work is about the same as a basecase division, but with more
|
||
function call overheads and with some subtractions separated from the
|
||
multiplies. These overheads mean that it's only when N/2 is above
|
||
`MUL_KARATSUBA_THRESHOLD' that divide and conquer is of use.
|
||
|
||
`DIV_DC_THRESHOLD' is based on the divisor size N, so it will be
|
||
somewhere above twice `MUL_KARATSUBA_THRESHOLD', but how much above
|
||
depends on the CPU. An optimized `mpn_mul_basecase' can lower
|
||
`DIV_DC_THRESHOLD' a little by offering a ready-made advantage over
|
||
repeated `mpn_submul_1' calls.
|
||
|
||
Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is
|
||
the time for an NxN multiplication done with FFTs. The actual time is
|
||
a sum over multiplications of the recursed sizes, as can be seen near
|
||
the end of section 2.2 of Burnikel and Ziegler. For example, within
|
||
the Toom-3 range, divide and conquer is 2.63*M(N). With higher
|
||
algorithms the M(N) term improves and the multiplier tends to log(N).
|
||
In practice, at moderate to large sizes, a 2NxN division is about 2 to
|
||
4 times slower than an NxN multiplication.
|
||
|
||
Newton's method used for division is asymptotically O(M(N)) and
|
||
should therefore be superior to divide and conquer, but it's believed
|
||
this would only be for large to very large N.
|
||
|
||
|
||
File: mpir.info, Node: Exact Division, Next: Exact Remainder, Prev: Divide and Conquer Division, Up: Division Algorithms
|
||
|
||
16.2.4 Exact Division
|
||
---------------------
|
||
|
||
A so-called exact division is when the dividend is known to be an exact
|
||
multiple of the divisor. Jebelean's exact division algorithm uses this
|
||
knowledge to make some significant optimizations (*note References::).
|
||
|
||
The idea can be illustrated in decimal for example with 368154
|
||
divided by 543. Because the low digit of the dividend is 4, the low
|
||
digit of the quotient must be 8. This is arrived at from 4*7 mod 10,
|
||
using the fact 7 is the modular inverse of 3 (the low digit of the
|
||
divisor), since 3*7 == 1 mod 10. So 8*543=4344 can be subtracted from
|
||
the dividend leaving 363810. Notice the low digit has become zero.
|
||
|
||
The procedure is repeated at the second digit, with the next
|
||
quotient digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving
|
||
325800. And finally at the third digit with quotient digit 6 (8*7 mod
|
||
10), subtracting 6*543=3258 leaving 0. So the quotient is 678.
|
||
|
||
Notice however that the multiplies and subtractions don't need to
|
||
extend past the low three digits of the dividend, since that's enough
|
||
to determine the three quotient digits. For the last quotient digit no
|
||
subtraction is needed at all. On a 2NxN division like this one, only
|
||
about half the work of a normal basecase division is necessary.
|
||
|
||
For an NxM exact division producing Q=N-M quotient limbs, the saving
|
||
over a normal basecase division is in two parts. Firstly, each of the
|
||
Q quotient limbs needs only one multiply, not a 2x1 divide and
|
||
multiply. Secondly, the crossproducts are reduced when Q>M to
|
||
Q*M-M*(M+1)/2, or when Q<=M to Q*(Q-1)/2. Notice the savings are
|
||
complementary. If Q is big then many divisions are saved, or if Q is
|
||
small then the crossproducts reduce to a small number.
|
||
|
||
The modular inverse used is calculated efficiently by
|
||
`modlimb_invert' in `gmp-impl.h'. This does four multiplies for a
|
||
32-bit limb, or six for a 64-bit limb. `tune/modlinv.c' has some
|
||
alternate implementations that might suit processors better at bit
|
||
twiddling than multiplying.
|
||
|
||
The sub-quadratic exact division described by Jebelean in "Exact
|
||
Division with Karatsuba Complexity" is not currently implemented. It
|
||
uses a rearrangement similar to the divide and conquer for normal
|
||
division (*note Divide and Conquer Division::), but operating from low
|
||
to high. A further possibility not currently implemented is
|
||
"Bidirectional Exact Integer Division" by Krandick and Jebelean which
|
||
forms quotient limbs from both the high and low ends of the dividend,
|
||
and can halve once more the number of crossproducts needed in a 2NxN
|
||
division.
|
||
|
||
A special case exact division by 3 exists in `mpn_divexact_by3',
|
||
supporting Toom-3 multiplication and `mpq' canonicalizations. It forms
|
||
quotient digits with a multiply by the modular inverse of 3 (which is
|
||
`0xAA..AAB') and uses two comparisons to determine a borrow for the next
|
||
limb. The multiplications don't need to be on the dependent chain, as
|
||
long as the effect of the borrows is applied, which can help chips with
|
||
pipelined multipliers.
|
||
|
||
|
||
File: mpir.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
|
||
|
||
16.2.5 Exact Remainder
|
||
----------------------
|
||
|
||
If the exact division algorithm is done with a full subtraction at each
|
||
stage and the dividend isn't a multiple of the divisor, then low zero
|
||
limbs are produced but with a remainder in the high limbs. For
|
||
dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this
|
||
remainder r is of the form
|
||
|
||
a = q*d + r*b^n
|
||
|
||
n represents the number of zero limbs produced by the subtractions,
|
||
that being the number of limbs produced for q. r will be in the range
|
||
0<=r<d and can be viewed as a remainder, but one shifted up by a factor
|
||
of b^n.
|
||
|
||
Carrying out full subtractions at each stage means the same number
|
||
of cross products must be done as a normal division, but there's still
|
||
some single limb divisions saved. When d is a single limb some
|
||
simplifications arise, providing good speedups on a number of
|
||
processors.
|
||
|
||
`mpn_bdivmod', `mpn_divexact_by3', `mpn_modexact_1_odd' and the
|
||
`redc' function in `mpz_powm' differ subtly in how they return r,
|
||
leading to some negations in the above formula, but all are essentially
|
||
the same.
|
||
|
||
Clearly r is zero when a is a multiple of d, and this leads to
|
||
divisibility or congruence tests which are potentially more efficient
|
||
than a normal division.
|
||
|
||
The factor of b^n on r can be ignored in a GCD when d is odd, hence
|
||
the use of `mpn_bdivmod' in `mpn_gcd', and the use of
|
||
`mpn_modexact_1_odd' by `mpn_gcd_1' and `mpz_kronecker_ui' etc (*note
|
||
Greatest Common Divisor Algorithms::).
|
||
|
||
Montgomery's REDC method for modular multiplications uses operands
|
||
of the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n)
|
||
uses the factor of b^n in the exact remainder to reach a product in the
|
||
same form (x*y)*b^-n (*note Modular Powering Algorithm::).
|
||
|
||
Notice that r generally gives no useful information about the
|
||
ordinary remainder a mod d since b^n mod d could be anything. If
|
||
however b^n == 1 mod d, then r is the negative of the ordinary
|
||
remainder. This occurs whenever d is a factor of b^n-1, as for example
|
||
with 3 in `mpn_divexact_by3'. For a 32 or 64 bit limb other such
|
||
factors include 5, 17 and 257, but no particular use has been found for
|
||
this.
|
||
|
||
|
||
File: mpir.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
|
||
|
||
16.2.6 Small Quotient Division
|
||
------------------------------
|
||
|
||
An NxM division where the number of quotient limbs Q=N-M is small can
|
||
be optimized somewhat.
|
||
|
||
An ordinary basecase division normalizes the divisor by shifting it
|
||
to make the high bit set, shifting the dividend accordingly, and
|
||
shifting the remainder back down at the end of the calculation. This
|
||
is wasteful if only a few quotient limbs are to be formed. Instead a
|
||
division of just the top 2*Q limbs of the dividend by the top Q limbs
|
||
of the divisor can be used to form a trial quotient. This requires
|
||
only those limbs normalized, not the whole of the divisor and dividend.
|
||
|
||
A multiply and subtract then applies the trial quotient to the M-Q
|
||
unused limbs of the divisor and N-Q dividend limbs (which includes Q
|
||
limbs remaining from the trial quotient division). The starting trial
|
||
quotient can be 1 or 2 too big, but all cases of 2 too big and most
|
||
cases of 1 too big are detected by first comparing the most significant
|
||
limbs that will arise from the subtraction. An addback is done if the
|
||
quotient still turns out to be 1 too big.
|
||
|
||
This whole procedure is essentially the same as one step of the
|
||
basecase algorithm done in a Q limb base, though with the trial
|
||
quotient test done only with the high limbs, not an entire Q limb
|
||
"digit" product. The correctness of this weaker test can be
|
||
established by following the argument of Knuth section 4.3.1 exercise
|
||
20 but with the v2*q>b*r+u2 condition appropriately relaxed.
|
||
|
||
|
||
File: mpir.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
|
||
|
||
16.3 Greatest Common Divisor
|
||
============================
|
||
|
||
* Menu:
|
||
|
||
* Binary GCD::
|
||
* Accelerated GCD::
|
||
* Extended GCD::
|
||
* Jacobi Symbol::
|
||
|
||
|
||
File: mpir.info, Node: Binary GCD, Next: Accelerated GCD, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
|
||
|
||
16.3.1 Binary GCD
|
||
-----------------
|
||
|
||
At small sizes MPIR uses an O(N^2) binary style GCD. This is described
|
||
in many textbooks, for example Knuth section 4.5.2 algorithm B. It
|
||
simply consists of successively reducing odd operands a and b using
|
||
|
||
a,b = abs(a-b),min(a,b)
|
||
strip factors of 2 from a
|
||
|
||
The Euclidean GCD algorithm, as per Knuth algorithms E and A,
|
||
reduces using a mod b but this has so far been found to be slower
|
||
everywhere. One reason the binary method does well is that the implied
|
||
quotient at each step is usually small, so often only one or two
|
||
subtractions are needed to get the same effect as a division.
|
||
Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth
|
||
section 4.5.3 Theorem E.
|
||
|
||
When the implied quotient is large, meaning b is much smaller than
|
||
a, then a division is worthwhile. This is the basis for the initial a
|
||
mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1
|
||
and 1x1 cases). But after that initial reduction, big quotients occur
|
||
too rarely to make it worth checking for them.
|
||
|
||
|
||
The final 1x1 GCD in `mpn_gcd_1' is done in the generic C code as
|
||
described above. For two N-bit operands, the algorithm takes about
|
||
0.68 iterations per bit. For optimum performance some attention needs
|
||
to be paid to the way the factors of 2 are stripped from a.
|
||
|
||
Firstly it may be noted that in twos complement the number of low
|
||
zero bits on a-b is the same as b-a, so counting or testing can begin on
|
||
a-b without waiting for abs(a-b) to be determined.
|
||
|
||
A loop stripping low zero bits tends not to branch predict well,
|
||
since the condition is data dependent. But on average there's only a
|
||
few low zeros, so an option is to strip one or two bits arithmetically
|
||
then loop for more (as done for AMD K6). Or use a lookup table to get
|
||
a count for several bits then loop for more (as done for AMD K7). An
|
||
alternative approach is to keep just one of a or b odd and iterate
|
||
|
||
a,b = abs(a-b), min(a,b)
|
||
a = a/2 if even
|
||
b = b/2 if even
|
||
|
||
This requires about 1.25 iterations per bit, but stripping of a
|
||
single bit at each step avoids any branching. Repeating the bit strip
|
||
reduces to about 0.9 iterations per bit, which may be a worthwhile
|
||
tradeoff.
|
||
|
||
Generally with the above approaches a speed of perhaps 6 cycles per
|
||
bit can be achieved, which is still not terribly fast with for instance
|
||
a 64-bit GCD taking nearly 400 cycles. It's this sort of time which
|
||
means it's not usually advantageous to combine a set of divisibility
|
||
tests into a GCD.
|
||
|
||
|
||
File: mpir.info, Node: Accelerated GCD, Next: Extended GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
|
||
|
||
16.3.2 Accelerated GCD
|
||
----------------------
|
||
|
||
For sizes above `GCD_ACCEL_THRESHOLD', MPIR uses the Accelerated GCD
|
||
algorithm described independently by Weber and Jebelean (the latter as
|
||
the "Generalized Binary" algorithm), *note References::. This
|
||
algorithm is still O(N^2), but is much faster than the binary algorithm
|
||
since it does fewer multi-precision operations. It consists of
|
||
alternating the k-ary reduction by Sorenson, and a "dmod" exact
|
||
remainder reduction.
|
||
|
||
For operands u and v the k-ary reduction replaces u with n*v-d*u
|
||
where n and d are single limb values chosen to give two trailing zero
|
||
limbs on that value, which can be stripped. n and d are calculated
|
||
using an algorithm similar to half of a two limb GCD (see `find_a' in
|
||
`mpn/generic/gcd.c').
|
||
|
||
When u and v differ in size by more than a certain number of bits, a
|
||
dmod is performed to zero out bits at the low end of the larger. It
|
||
consists of an exact remainder style division applied to an appropriate
|
||
number of bits (*note Exact Division::, and *note Exact Remainder::).
|
||
This is faster than a k-ary reduction but useful only when the operands
|
||
differ in size. There's a dmod after each k-ary reduction, and if the
|
||
dmod leaves the operands still differing in size then it's repeated.
|
||
|
||
The k-ary reduction step can introduce spurious factors into the GCD
|
||
calculated, and these are eliminated at the end by taking GCDs with the
|
||
original inputs gcd(u,gcd(v,g)) using the binary algorithm. Since g is
|
||
almost always small this takes very little time.
|
||
|
||
At small sizes the algorithm needs a good implementation of
|
||
`find_a'. At larger sizes it's dominated by `mpn_addmul_1' applying n
|
||
and d.
|
||
|
||
|
||
File: mpir.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Accelerated GCD, Up: Greatest Common Divisor Algorithms
|
||
|
||
16.3.3 Extended GCD
|
||
-------------------
|
||
|
||
The extended GCD calculates gcd(a,b) and also cofactors x and y
|
||
satisfying a*x+b*y=gcd(a,b). Lehmer's multi-step improvement of the
|
||
extended Euclidean algorithm is used. See Knuth section 4.5.2
|
||
algorithm L, and `mpn/generic/gcdext.c'. This is an O(N^2) algorithm.
|
||
|
||
The multipliers at each step are found using single limb
|
||
calculations for sizes up to `GCDEXT_THRESHOLD', or double limb
|
||
calculations above that. The single limb code is faster but doesn't
|
||
produce full-limb multipliers, hence not making full use of the
|
||
`mpn_addmul_1' calls.
|
||
|
||
When a CPU has a data-dependent multiplier, meaning one which is
|
||
faster on operands with fewer bits, the extra work in the double-limb
|
||
calculation might only save some looping overheads, leading to a large
|
||
`GCDEXT_THRESHOLD'.
|
||
|
||
Currently the single limb calculation doesn't optimize for the small
|
||
quotients that often occur, and this can lead to unusually low values of
|
||
`GCDEXT_THRESHOLD', depending on the CPU.
|
||
|
||
An analysis of double-limb calculations can be found in "A
|
||
Double-Digit Lehmer-Euclid Algorithm" by Jebelean (*note References::).
|
||
The code in MPIR was developed independently.
|
||
|
||
It should be noted that when a double limb calculation is used, it's
|
||
used for the whole of that GCD, it doesn't fall back to single limb
|
||
part way through. This is because as the algorithm proceeds, the
|
||
inputs a and b are reduced, but the cofactors x and y grow, so the
|
||
multipliers at each step are applied to a roughly constant total number
|
||
of limbs.
|
||
|
||
|
||
File: mpir.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
|
||
|
||
16.3.4 Jacobi Symbol
|
||
--------------------
|
||
|
||
`mpz_jacobi' and `mpz_kronecker' are currently implemented with a
|
||
simple binary algorithm similar to that described for the GCDs (*note
|
||
Binary GCD::). They're not very fast when both inputs are large.
|
||
Lehmer's multi-step improvement or a binary based multi-step algorithm
|
||
is likely to be better.
|
||
|
||
When one operand fits a single limb, and that includes
|
||
`mpz_kronecker_ui' and friends, an initial reduction is done with
|
||
either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary
|
||
algorithm on a single limb. The binary algorithm is well suited to a
|
||
single limb, and the whole calculation in this case is quite efficient.
|
||
|
||
In all the routines sign changes for the result are accumulated
|
||
using some bit twiddling, avoiding table lookups or conditional jumps.
|
||
|
||
|
||
File: mpir.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
|
||
|
||
16.4 Powering Algorithms
|
||
========================
|
||
|
||
* Menu:
|
||
|
||
* Normal Powering Algorithm::
|
||
* Modular Powering Algorithm::
|
||
|
||
|
||
File: mpir.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
|
||
|
||
16.4.1 Normal Powering
|
||
----------------------
|
||
|
||
Normal `mpz' or `mpf' powering uses a simple binary algorithm,
|
||
successively squaring and then multiplying by the base when a 1 bit is
|
||
seen in the exponent, as per Knuth section 4.6.3. The "left to right"
|
||
variant described there is used rather than algorithm A, since it's
|
||
just as easy and can be done with somewhat less temporary memory.
|
||
|
||
|
||
File: mpir.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms
|
||
|
||
16.4.2 Modular Powering
|
||
-----------------------
|
||
|
||
Modular powering is implemented using a 2^k-ary sliding window
|
||
algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
|
||
(*note References::). k is chosen according to the size of the
|
||
exponent. Larger exponents use larger values of k, the choice being
|
||
made to minimize the average number of multiplications that must
|
||
supplement the squaring.
|
||
|
||
The modular multiplies and squares use either a simple division or
|
||
the REDC method by Montgomery (*note References::). REDC is a little
|
||
faster, essentially saving N single limb divisions in a fashion similar
|
||
to an exact remainder (*note Exact Remainder::). The current REDC has
|
||
some limitations. It's only O(N^2) so above `POWM_THRESHOLD' division
|
||
becomes faster and is used. It doesn't attempt to detect small bases,
|
||
but rather always uses a REDC form, which is usually a full size
|
||
operand. And lastly it's only applied to odd moduli.
|
||
|
||
|
||
File: mpir.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms
|
||
|
||
16.5 Root Extraction Algorithms
|
||
===============================
|
||
|
||
* Menu:
|
||
|
||
* Square Root Algorithm::
|
||
* Nth Root Algorithm::
|
||
* Perfect Square Algorithm::
|
||
* Perfect Power Algorithm::
|
||
|
||
|
||
File: mpir.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms
|
||
|
||
16.5.1 Square Root
|
||
------------------
|
||
|
||
Square roots are taken using the "Karatsuba Square Root" algorithm by
|
||
Paul Zimmermann (*note References::).
|
||
|
||
An input n is split into four parts of k bits each, so with b=2^k we
|
||
have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so
|
||
that either the high or second highest bit is set. In MPIR, k is kept
|
||
on a limb boundary and the input is left shifted (by an even number of
|
||
bits) to normalize.
|
||
|
||
The square root of the high two parts is taken, by recursive
|
||
application of the algorithm (bottoming out in a one-limb Newton's
|
||
method),
|
||
|
||
s1,r1 = sqrtrem (a3*b + a2)
|
||
|
||
This is an approximation to the desired root and is extended by a
|
||
division to give s,r,
|
||
|
||
q,u = divrem (r1*b + a1, 2*s1)
|
||
s = s1*b + q
|
||
r = u*b + a0 - q^2
|
||
|
||
The normalization requirement on a3 means at this point s is either
|
||
correct or 1 too big. r is negative in the latter case, so
|
||
|
||
if r < 0 then
|
||
r = r + 2*s - 1
|
||
s = s - 1
|
||
|
||
The algorithm is expressed in a divide and conquer form, but as
|
||
noted in the paper it can also be viewed as a discrete variant of
|
||
Newton's method, or as a variation on the schoolboy method (no longer
|
||
taught) for square roots two digits at a time.
|
||
|
||
If the remainder r is not required then usually only a few high limbs
|
||
of r and u need to be calculated to determine whether an adjustment to
|
||
s is required. This optimization is not currently implemented.
|
||
|
||
In the Karatsuba multiplication range this algorithm is
|
||
O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
|
||
limbs. In the FFT multiplication range this grows to a bound of
|
||
O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the
|
||
Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
|
||
|
||
The algorithm does all its calculations in integers and the resulting
|
||
`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended
|
||
precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.
|
||
|
||
|
||
File: mpir.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms
|
||
|
||
16.5.2 Nth Root
|
||
---------------
|
||
|
||
Integer Nth roots are taken using Newton's method with the following
|
||
iteration, where A is the input and n is the root to be taken.
|
||
|
||
1 A
|
||
a[i+1] = - * ( --------- + (n-1)*a[i] )
|
||
n a[i]^(n-1)
|
||
|
||
The initial approximation a[1] is generated bitwise by successively
|
||
powering a trial root with or without new 1 bits, aiming to be just
|
||
above the true root. The iteration converges quadratically when
|
||
started from a good approximation. When n is large more initial bits
|
||
are needed to get good convergence. The current implementation is not
|
||
particularly well optimized.
|
||
|
||
|
||
File: mpir.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms
|
||
|
||
16.5.3 Perfect Square
|
||
---------------------
|
||
|
||
A significant fraction of non-squares can be quickly identified by
|
||
checking whether the input is a quadratic residue modulo small integers.
|
||
|
||
`mpz_perfect_square_p' first tests the input mod 256, which means
|
||
just examining the low byte. Only 44 different values occur for
|
||
squares mod 256, so 82.8% of inputs can be immediately identified as
|
||
non-squares.
|
||
|
||
On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17,
|
||
for a total 99.25% of inputs identified as non-squares. On a 64-bit
|
||
system 97 is tested too, for a total 99.62%.
|
||
|
||
These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
|
||
for 64-bits), and such a remainder can be quickly taken just using
|
||
additions (see `mpn_mod_34lsub1').
|
||
|
||
When nails are in use moduli are instead selected by the `gen-psqr.c'
|
||
program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1
|
||
could be done with nails using some extra bit shifts, but this is not
|
||
currently implemented.
|
||
|
||
In any case each modulus is applied to the `mpn_mod_34lsub1' or
|
||
`mpn_mod_1' remainder and a table lookup identifies non-squares. By
|
||
using a "modexact" style calculation, and suitably permuted tables,
|
||
just one multiply each is required, see the code for details. Moduli
|
||
are also combined to save operations, so long as the lookup tables
|
||
don't become too big. `gen-psqr.c' does all the pre-calculations.
|
||
|
||
A square root must still be taken for any value that passes these
|
||
tests, to verify it's really a square and not one of the small fraction
|
||
of non-squares that get through (ie. a pseudo-square to all the tested
|
||
bases).
|
||
|
||
Clearly more residue tests could be done, `mpz_perfect_square_p' only
|
||
uses a compact and efficient set. Big inputs would probably benefit
|
||
from more residue testing, small inputs might be better off with less.
|
||
The assumed distribution of squares versus non-squares in the input
|
||
would affect such considerations.
|
||
|
||
|
||
File: mpir.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
|
||
|
||
16.5.4 Perfect Power
|
||
--------------------
|
||
|
||
Detecting perfect powers is required by some factorization algorithms.
|
||
Currently `mpz_perfect_power_p' is implemented using repeated Nth root
|
||
extractions, though naturally only prime roots need to be considered.
|
||
(*Note Nth Root Algorithm::.)
|
||
|
||
If a prime divisor p with multiplicity e can be found, then only
|
||
roots which are divisors of e need to be considered, much reducing the
|
||
work necessary. To this end divisibility by a set of small primes is
|
||
checked.
|
||
|
||
|
||
File: mpir.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
|
||
|
||
16.6 Radix Conversion
|
||
=====================
|
||
|
||
Radix conversions are less important than other algorithms. A program
|
||
dominated by conversions should probably use a different data
|
||
representation.
|
||
|
||
* Menu:
|
||
|
||
* Binary to Radix::
|
||
* Radix to Binary::
|
||
|